Discoveries

• The ledger of MrlyMath: every claim on one line with its tag and its witness.
• Tags: Proved, Verified, Conjecture, Refuted. A witness is a crate function, a lab study, a shelf paper, or an OEIS entry.
• SETTLED holds Proved and Verified claims, OPEN holds Conjectures with the Proved and Verified steps beside them, REFUTED holds what fell.

SETTLED

Design census

• [Proved] Designs up to cube symmetry are the NP-equivalence classes of Boolean functions: an equivariant bijection carries B_D (order 2^D * D!) onto the NP group, so the class counts are 3, 6, 22, 402, 1228158, 400507806843728 at D = 1..6, reproduced by orbit walk on designs, orbit walk on truth tables and Burnside, checked against the entry to D = 7; the NPN sibling gives 2, 4, 14, 222 at D = 1..4. Witness: mrlymath::bang::counting::sequence, A000616, A000370.
• [Verified] The base-q design census at D = 2 is the toroidal binary array count: Burnside over one dihedral group per residue axis gives 2, 6, 26, 805, 172112, 239123150, 1436120190288, 36028817512382026 at q = 1..8, with brute-force orbit closure agreeing at q = 3, 4. Witness: mrlymath::bang::baseq::distinct_designs, A255016.
• [Proved] The isotropic-class count is A005418(D+2) minus one at even D: a level set's orbit is {F_S xor t} and the image depends on t only through |t|, so the count is subsets of {0..D} up to reversal with the even-D merge, 3, 5, 10, 19, 36, 71, 136, 271, 528, 1055, 2080, 4159, 8256, 16511, 32896, 65791 at D = 1..16; nameable classes grow like 2^D, half the 2^(D+1) subsets. Witness: A005418.
• [Verified] Census multiplicity in the base-2 census is a bounded perfect-power representation count: over the 16 two-dimensional and 256 three-dimensional designs at side 3 and levels 1..5, the 1360 design-level pairs give 119 distinct fill counts, M(N) = sum_{L=1}^5 sum_{b=0}^27 c_b [b^L = N] with sum_b c_b x^b = prod_{w in {1,2,4}} (1 + x^w) + prod_{w in {1,2,4,8}} (1 + x^w), maximum M(4096) = 29 from 4096 = 8^4 = 16^3 (14 designs of base fill 8 plus 15 of base fill 16), support exactly {b^L : 0 <= b <= 27, 1 <= L <= 5}, so only 119 integers occur up to 27^5 = 14348907, coverage 8.29e-6, in 91 maximal missing runs, the longest 11881377..14348906; M is not multiplicative (M(2) = M(3) = 5, M(6) = 12), opens 10, 10, 5, 5, 13, 8, 12, 12, 19, 19, 15, 15, 16, 16, 16, 16, and no classical arithmetic function is behind it: the raw Pearson signals against sigma and phi (-0.332, -0.315) are shared size dependence (M against N is -0.338), and partial rank correlations controlling for log N fall in -0.035..0.096 for d, sigma, phi, omega, Omega. Witness: mrlymath::formulas::counting::fill.
• [Verified] The census covers only a short prefix of the integers: through Kronecker level 6 the 256 three-dimensional designs produce 37 positive fill values in 1..262144, coverage 0.01411%, contiguous only on 1..9, because fill(code, L) = f^L with f the tile popcount in 0..8; the nine named observables reach a union of 368 integers to 1633932 with 10 the first gap, the graph observables core_edges, tips, junctions extend it to 437 integers and the prefix to 1..24; edges is the broadest single observable (73 distinct positive values, 60 exclusive), then faces 70, vertices 67, surface 66; design 23 reads fill 20, voids 7, surface 72, vertices 64, edges 144, faces 96, Euler -4 at level 1 and fill 400, vertices 896, edges 2304, faces 1728, Euler -80 at level 2, and every record obeys surface = 6 fill - 2 core_edges, faces = 6 fill - core_edges, cycle_rank = core_edges - fill + components, euler = vertices - edges + faces - fill; voids means every empty lattice site, 3D edges means cubical-complex unit edges and core_edges the branch count of the face-adjacency graph; the 2D census gives 176 distinct positive integers to 1064340, contiguous on 1..10. Witness: mrlymath::formulas::counting::fill, mrlymath::three::census::census.
• [Verified] Unbounded census observables cannot be automatic sequences and are at best k-regular: an integer-valued k-automatic sequence has finite range, a finite-range k-regular sequence is k-automatic, so d(n) and sigma(n) are k-automatic in no base and can enter only through unbounded k-regular representations, weighted substitutions, Dirichlet convolutions or a purpose-built geometric model; Cobham's theorem is the one general rigidity constraint, and substitution incidence matrices generate additive recurrences while d and sigma are multiplicative over primes, so self-similar census counts c_m = u^T A^m v are generically sparse in the integers. Witness: REFS.md.

Diagonal cuts of the parity solid

• [Proved] Every diagonal slice of mrly_bang_d3_126 holds exactly 3^L points at every admissible height, by uniqueness of the binary expansion of the height offset; checked at L = 1..8 by two enumerations sharing no code and to L = 14 by a height recursion; the constancy separates it from the digit-scheduled slices of Nakajima and Watanabe, whose non-autonomous IFS uses A_c^(j) = {0} or {1,2,3} by the height digit, so their digit changes the number of maps and this one only the orientation. Witness: mrlymath::three::diagonal.
• [Verified] The two central cuts of mrly_bang_d3_126 decompose into six congruent Sierpinski gaskets of 3^(L-1) points each tiling a hexagon with an order-12 symmetry group, checked at L = 2..8 by rebuilding the pieces from level L-1 (union, pairwise disjointness, sizes); the combined totals are 6 * 3^(L-1) = 2 * 3^L: 18, 54, 162, 486, 1458, 4374, 13122; the ambient object is the octahedron flake of dimension log(6)/log(2). Witness: mrlymath::three::diagonal.

Coprimality density

• [Proved] The base-local coprimality factor is exact at every finite level: #{x in S_n : e | x_i for all i} = k_e * k^(n-1) for every squarefree e | rad(q), holding on all 763 census lines; at composite base it does not factor over primes, the base-6 sample giving B(F) = 1/2 (256 and 240 of 512) against the naive 0.46875; the character contraction c(q,k) = 1 - (2/k)(1 - cos(pi/(2q))) gives 0.804738, 0.966506, 0.946410, 0.986603, 0.991481 at (q,k) = (2,3), (3,8), (3,5), (3,20), (6,8). Witness: coprime-density-above-dimension-one, lab/design-census.
• [Verified] Lemma B, the equidistribution estimate the coprimality programme needs, is a uniform equidistribution statement for digit-restricted sets across moduli growing with the level, the subject of Erdos, Mauduit and Sarkozy 1998, Konyagin 2001 and Maynard 2019, with Lemma A supplying the per-character input; it is proved for every design with k > q by the dimension-above-one theorem and remains open only at k <= q. Witness: coprime-density-above-dimension-one, REFS.md.
• [Proved] Every digital design of fractal dimension above one has the classical coprimality density: for D >= 2, condition (E) and k > q, A(n)/k^n -> B(F) prod_{p not dividing q} (1 - p^(-D)), by the box bound N*_n(m) <= (q+1)^D k^n m^(-log_q k) driving a Chebyshev log-gcd sum and a fixed-z sieve; this settles the gasket 16/(3 pi^2), the or-triangle 8/pi^2, the carpet 189/(32 pi^2), the Vicsek plus 27/(4 pi^2) and the sponge (513/520)/zeta(3), and the finite levels approach with oscillating signed error: carpet gap -3.52e-07 at n = 20, sponge -3.45e-05 at n = 18 (lab/dimension-one-ladder), gasket 0.539591 against 0.540380 at n = 16 (lab/oeis-terms, A396934). Witness: coprime-density-above-dimension-one, lab/dimension-one-ladder, lab/oeis-terms, A396934.
• [Proved] The 36 open dimension-one census lines are one problem: for any full-rank q = 3, k = 3 design, collecting the corner-choice classes into E_j = sum_{c_l = j} 3^l maps S_n bijectively through the gasket, and for every m coprime to the design's difference determinant the two divisibility conditions become one shifted-target congruence on the gasket pair, so Lemma B stands or falls for the whole family at once, with T* vanishing at m >= 3^n; the same argument reduces every full-rank k = D + 1 design at any base to the base-q simplex. Witness: lab/design-census.
• [Verified] Not every census collision is a shear: base-3 codes 11 and 161 have identical A(n) at every level (2, 4, 12, 34, 108, 322, 992, 3006, 8924, ... through n = 12, and equal T_e for every e <= 40 at n <= 7) though 161 is alone in its GL_2(Z) orbit over the exhaustive entry range [-8, 8]: both have zero corner v_0 = 0 hence gasket target 0, determinants with prime support {3}, and matching base-3 peel, so the finite Mobius sums agree termwise. Witness: lab/design-census.
• [Proved] A(n) is not C-finite for any design meeting the dimension-above-one theorem with B(F) > 0 and D even or D = 3: a rational C-finite sequence with A(n)/k^n convergent has a rational limit (roots above k have zero coefficient, oscillatory roots on |z| = k die by mean-square averaging, the remaining constant is fixed by every Galois automorphism), while delta is an irrational multiple of 1/zeta(D); all five eligible base-2 plane designs through n = 12 admit no rational constant-coefficient recurrence of order at most 6 and approach their irrational limits (the gasket-type designs read 0.5378546 at level 12 against 0.5403796); the theorem says nothing at odd D >= 5, at B(F) = 0 or k <= q, or about polynomial-coefficient recurrences, which 2729 exact P-recursive fits with held-out terms exclude only empirically. Witness: coprime-density-above-dimension-one.
• [Proved] The Menger sponge's pairwise coprimality density is (13/20) prod_{p != 3} (1 - 3/p^2 + 2/p^3) = 0.251620868451255 = (351/400) C_3 with C_3 = 0.286747428434479, so the sponge rule lowers the full-lattice benchmark by exactly 12.25%: the three-modulus Mobius inversion over the three coordinate pairs does not collapse to one modulus, the base factor 13/20 is exact at every level (13 of the 20 legal digit vectors have at most one zero), each foreign prime contributes (1 - 1/p)^2 (1 + 2/p), and the tail closes on the pair-fibred box bound with kappa_I = 3 and alpha = log_3(20/3) = 1.726833 > 1; the exact census 0, 60, 1434, 32268, 721524, 15141288 at L = 1..6 gives 0, 0.150000, 0.179250, 0.201675, 0.225476, 0.236583; the local factor is not 1 - p^(-s), so no reciprocal zeta value is claimed and the coefficient in 0.4138997384/zeta(2) carries no rationality claim. Witness: menger-pairwise-coprimality.
• [Proved] The density theorem's spanning hypothesis retires to condition (E): a finite abelian quotient of order m is killed by m, so m Z^3 sits inside the difference lattice, and a character mod d vanishing on F - F forces m t = 0 mod d with gcd(m, d) = 1, hence t = 0, the only step of Lemma A that used spanning; non-parity index-4 and index-8 designs at bases 4 and 6 measure 0.105072, 0.035346, 0.936067 against the widened predictions 0.105639, 0.035261, 0.950751, converging; those three non-parity measurements have no generator in lab/, which computes parity codes only. Witness: coprime-density-above-dimension-one.
• [Proved] Every parity code at every even base q >= 4 has delta * zeta(3) = (8/7)(1 - W_0/|P|), nine values only and independent of q, because multiples of an odd m | q split evenly by parity while multiples of 2m are all even, so every odd base prime cancels its own Euler correction exactly; all 255 nonempty codes at every even q <= 40 give exactly nine band values, every lattice index at q = 4, 6, 8, 10 lies in {1, 2, 4, 8}, enumerations reproduce 0.712853, 0.951771, 0.709137, and in D = 2 the parity carpet's band value 8/9 gives 8/9 / zeta(2) = 0.5403796460924681 = 16/(3 pi^2), an even-base band constant rather than the gasket's own. Witness: lab/mrlybang-density-classes.
• [Proved] Odd bases are self-similar across bases and the density trichotomy is exhaustive for odd q >= 5: k_e(q) = k_P(q/e) for squarefree e | q, so the bracket is a Mobius convolution of the code's corner-count cubic; the 149 spanning codes converge to 1/zeta(3) along the odds while frozen on their even band, the 43 codes inside their difference span take the corrected factor 1 - 2^(-s2) at 2, and the 63 codes whose affine span avoids the origin have no density at all, with zero all-even points at every odd level and two subsequential limits in ratio 1 - 2^(-s2); the even value equals the odd limit exactly on the 16 subgroup codes; checked for all codes at all odd q <= 75, twelve measured cases to the printed digit including exact zeros at the even levels of {111} at q = 5 and the axes pair 0.987338 / 0.739563 against 0.987319 / 0.740489. Witness: lab/mrlybang-density-classes.
• [Proved] Slice coprimality is finite arithmetic of the height: on x + y + z = s the gcd divides s, so A_s = sum_{d | s} mu(d) N_s^(d) exactly with no tail and no Lemma B; prime slices are fully visible up to the three axis points (coordinates forced into {0, p}), the base prime peels the slice to level n - 1 one step off-centre, a code without the origin corner owes nothing at its base on any slice, and each foreign prime costs the slice 1/p^2 where it costs the solid 1/p^3 (aggregated locals 0.040902 against 1/25, 0.020446 against 1/49); the parity-walk factor at 2 is 9121792/32002048 on the integer, the net's immunity at 3 holds on all 7^7 points, and the tree dichotomy holds: even slices hold zero visible points, odd slices zero even gcds. Witness: lab/slice-coprimality.
• [Proved] The central slice never converges and its bill is the repunit: s* = (3(q-1)/2) R_n(q) owes 3 always, 2 exactly when q = 1 mod 4 or n is even, and a foreign odd prime exactly when ord_p(q) | n, so the centre's visible density is a quasiperiodic function of the divisors of n; at q = 3 the two streams read 0.892, 0.898 against 0.571, 0.611, 0.652, at n = 7 the whole foreign bill is the Wieferich prime 1093 (2^1092 = 1 mod 1093^2), the central count is A299916(n) on the (9, -12) recurrence exact to n = 14, and the sixth peeled term is 83835 by a meet-in-the-middle count over all 20^7 level-7 points. Witness: lab/slice-coprimality, A299916.

Boolean complexity

• [Verified] Geometry under-determines Boolean complexity at D = 4: mrly_bang_d4_27 and mrly_bang_d4_281 share genus, GF(2) degree, popcount and the fill polynomial 4k^4 - 4k^3 + k^2 and split six of seven complexity measures; across the 424 classes (22 + 402) 92 groups hold two or more classes and 279 measure splits occur. Witness: lab/boolean-measures, mrlymath::bang::counting::sequence.
• [Verified] Sensitivity and block sensitivity separate exactly once in the D <= 4 catalog: s = bs on all 22 classes at D = 3 and on 401 of 402 at D = 4, the exception mrly_bang_d4_7128 with s = 2, bs = 3, orbit 24; C = bs on all 424 classes; exactly two classes meet deg = s^2, mrly_bang_d4_855 and mrly_bang_d4_1911, the second the AND-of-ORs that Huang 2019 names tight for s(f) >= sqrt(deg(f)). Witness: lab/boolean-measures.

The hexagonal slice

• [Verified] The carpet face-count law V(i) = 2 * 20^i + 4 * 8^i, visible faces 6, 72, 1056, 18048, 336384, is A332705 verbatim, the surface area of the stage-i Menger sponge, with the same closed form on the entry. Witness: mrlymath::formulas::surface::surface, A332705.
• [Proved] The carpet slice census 6, 42, 306, 2250, 16578 is A299916(L+1): sectioning the level-L sponge on x + y + z = 1.5 * 3^L, a surviving cube cuts a hexagon of 6 mesh triangles or a triangle of 1, refinement triples the plane offset, and the 20 surviving subcubes split by coordinate sum as 1, 3, 3, 6, 3, 3, 1, so H_(L+1) = 6 H_L + T_L and T_(L+1) = 6 H_L + 3 T_L, the hexagon-triangle substitution proved by exhaustion; the ledger 54 = 6*6 + 6*1 + 12 punches exactly one 12-triangle hexagram per hexagon and none per triangle, so hexagram holes of the n-th size number A299916(n) and the mesh census is one index up; the recurrence a(n) = 9 a(n-1) - 12 a(n-2) gives the slice dimension log((9 + sqrt(33))/2)/log(3) = 1.818410; the empty area at L = 0..5 resolves into 1, 6, 42 components of descending size, each with six radial maxima, sixfold symmetry to 0.001 and max/min radius 1.68 against the hexagram's sqrt(3). Witness: slice-recurrence-order, mrlymath::six::topology test the_carpet_slice_percolates_at_base_three, A299916.
• [Verified] The slice vertex count 12k^2 - 6k + 1 is A154105 at n = k - 1 and the centered hexagonal number A003215 at index 2k - 1 (3m(m+1) + 1 at m = 2k - 1), so a prime vertex count is a cuban prime, A002407; at k = 1..20 ten values are prime, 7, 37, 271, 397, 547, 919, 1657, 1951, 2269, 4219, and ten composite, 91, 169, 721, 1141, 1387, 2611, 2977, 3367, 3781, 4681. Witness: mrlymath::formulas::six::solid_slice_vertices, A154105, A003215, A002407.

Farey stack

• [Verified] The odd-carpet stack renders an RH-equivalent object and no route to a proof: the lit nodes are exactly the Farey fractions, each scale n contributing phi(n) new nodes, their discrepancy is the object of the Franel and Landau 1924 theorems, and the measured S2 * Q flattens near 0.656 (0.6560 at 2000, 0.6564 at 8000) with the local exponent walking to -1. Witness: mrlynum::lattice::new_nodes, lab/farey-discrepancy.
• [Proved] The stack is an address, not a construction: the odd-carpet stack's brightness at x = (a_1/q, a_2/q) is the residue count B_N(x) = ceil(N/2) - sum_{r in S(x), r <= N} (floor((N - r)/(2q)) + 1) over the bad residues mod 2q, per-point cost independent of N, and the line stack's is floor(N/b); a stack of 5 * 10^17 layers, N = 10^18, evaluates exactly in a tenth of a second, and at N = 55 the Farey table holds 940 nodes summing to 1540 = N(N+1)/2; the closed form's proved boundaries are per-point only (an R x R raster costs R^2), exact representations only (a real-oracle input is undecidable on {n x integer}, irrationals with known continued fractions stay computable via Ostrowski), finite N only (infinite-depth membership is undecidable) and unweighted only. Witness: lab/carpet-stack-address, mrlynum::lattice::farey.
• [Verified] Immediacy buys no RH content: the Mobius-weighted node is the Mertens-type sum Sum_{k <= N/b} mu(kb), equal to M(N) at b = 1 and to M(floor(N/b)) at only 64 of 200 denominators at N = 200, with no polynomial-time algorithm for the Mertens function at binary input and the best known near x^(2/3); the rank closed form sum_d mu(d) sum_e floor(x e) re-imports Mobius, the meter's global readout collapses to sum_{n <= N} M(floor(N/n)) = 1 identically (checked exactly to N = 20000), the divisibility incidence array is the Redheffer matrix up to its first column, and Franel 1924 is already the symbolic all-Q reduction, so the route ends at Mertens. Witness: lab/mertens-meter.
• [Verified] The stack's complexity frontier is the sharing of its scales: per-pixel brightness with binary inputs is in P by fixed-dimension lattice-point counting (Barvinok 1994, two parity branches summed), destroying the shared scales makes "does any point reach maximum brightness" NP-complete (Simultaneous Incongruences, Garey and Johnson SP3), and making the ambient dimension part of the input makes "is any layer lit at this fixed point" NP-complete (Lagarias 1985) while polynomial at every fixed dimension, so a no-shortcut theorem for this stack could never separate P from NP; d(n) is not factoring-hard by the sigma route (sigma(pq) = pq + p + q + 1 recovers the factors while d(pq) = 4 carries nothing), and the O(q) residue sweep is polynomial in q, hence a unary-input algorithm. Witness: REFS.md.

Walk dimension

• [Verified] Mass does not fix music: codes 127 and 239 share fill 7 and d_f = log(7)/log(3) = 1.771244 with identical density at every level, yet their walk dimensions separate by about 0.39 (2.26 against 2.64), both generators agreeing and every drift bar an order of magnitude smaller than the gap. Witness: lab/walk-dimension.

Flake band gap

• [Verified] The base-2 flake's Laplacian has an interior band gap whose upper edge is exactly 4: on the mrly_bang_d3_23 flake (4^L nodes, 4^L - 1 edges, connected), exact rational elimination of Lap - 4I has a single zero pivot, 3 * 4^(L-1) eigenvalues lie below 2 and none in [2, 4), the lower edge climbs 1.000000, 1.827520, 1.975680, 1.996862, 1.999605, 1.999950 at L = 1..6, and the top eigenvalue is 3 + sqrt(5) = 5.2360679775 at L = 2 climbing to 5.7090316570 at L = 6. Witness: lab/flake-band-gap.

Menger slice sources

• [Verified] Cook's 2011 slice code is a point-sampled raster, not an array-free predicate: the listing allocates an n x n integer array, fills it by sampling the digit predicate and renders it, so the exact-census method is this tree's and not Cook's; its prose puts the normal through (1, 1, 1) while the listing sets (1, 1, 0.5). Witness: REFS.md.
• [Verified] Hart 2012 is an exact citation for the diagonal Menger slice: the Simons Foundation page carries his byline, the title "Mathematical Impressions: The Surprising Menger Sponge Slice" and a photograph credit. Witness: REFS.md.
• [Verified] "The Marstrand value d - 1" is the wrong attribution for a plane cutting a solid in R^3: Marstrand 1954 concerns plane sets and lines in R^2, and the hyperplane generalisation is Mattila 1975; both are almost-everywhere statements, so one maximally arithmetic plane landing above or below d - 1 contradicts neither. Witness: REFS.md.
• [Verified] A299916's Menger reading is not its definition: the entry's name is a(n) = A299914(2n+1), offset 0, signature (9, -12), terms 1, 6, 42, 306, 2250, 16578, its reference is number theory with no sponge in it, and the hexagram-hole geometry lives in one comment with one uploaded picture, which Wikipedia sources onward beside a newspaper article; it must be cited as a comment, never as the sequence's definition. Witness: A299916.
• [Verified] Abel 2012 claims no proof of the base-3 slice dimension, only a computation, naming mass distributions and similarity graphs as routes to one, so every generalised dimension inherits that status. Witness: REFS.md.

Odd-base slice grammar

• [Verified] The odd-base generalisation rests on a choice of solid: Cook's predicate "at most one coordinate in the middle third" and the mrly_bang_d3_23 rule "at most one odd coordinate" agree at b = 3 (20 of 27) and nowhere else, at b = 5 filling 4^3 + 3 * 4^2 = 112 of 125 against 3^3 + 3 * 2 * 3^2 = 81 = 4k^3 - 3k^2 at k = 3, and there is no canonical base-5 Menger sponge. Witness: lab/odd-base-slice-grammar.
• [Verified] In the comparison of the slice dimension against d - 1, d is the solid's own log_b(fill) and never the ambient 3: with d = 3 the value d - 1 = 2 exceeds all four slice dimensions 1.8184, 1.6869, 1.8026, 1.7204 and the mod-4 split collapses. Witness: lab/odd-base-slice-grammar.
• [Verified] The four printed dimensions are consistent with the four printed rules and this is not evidence for either: the dominant roots (9 + sqrt(33))/2 = 7.37228, (11 + sqrt(369))/2 = 15.1047, (42 + sqrt(612))/2 = 33.3693, (28 + sqrt(3556))/2 = 43.8161 give log_b of 1.8183, 1.6870, 1.8026, 1.7204 at b = 3, 5, 7, 9, while 4k^3 - 3k^2 at k = 2..5 gives 20, 81, 208, 425 and d - 1 = 1.7268, 1.7304, 1.7430, 1.7544; a rule and its own dimension cannot cross-check each other. Witness: lab/odd-base-slice-grammar.
• [Proved] The middle diagonal layer sits at coordinate sum 3(b-1)/2, odd exactly when b = 3 mod 4 (3, 6, 9, 12 at b = 3, 5, 7, 9). Witness: lab/odd-base-slice-grammar.

Flat carpet stack

• [Proved] The moire correlation law: for odd m, n the mean of s(mu) s(nu) with s(x) = (-1)^floor(x) is exactly gcd(m,n)^2/(mn) (for general integers it needs m/g and n/g both odd, else the integral is 0), and the Pearson correlation of the 1D parity indicators is exactly (gcd^2 - 1)/sqrt((m^2 - 1)(n^2 - 1)), so the correlation is exactly 0 if and only if the scales are coprime, and zero covariance is independence for Bernoulli pictures; exact rational integration over all odd pairs to 99 matches the closed forms to 5.6e-17. Witness: moire-correlation-laws.
• [Proved] The stack is an exact prime detector: an odd n >= 3 is prime exactly when its carpet is uncorrelated with every earlier carpet; over odd 3..199 all 45 primes sit at exactly 0 and all 54 composites strictly positive, minimum 0.0517383 at n = 169 = 13^2; the finite-window corollary "the zero-redundancy layers of the 1..55 stack are the primes above 55/3" is a window artifact and not the statement. Witness: moire-correlation-laws.
• [Proved] Pi cancels out of every visible brightness of the stack: ray strengths, crosshair steps, hot-spot values, layer correlations and per-layer means are all rational, because the square wave's (4/pi)^2 meets the odd Basel sum pi^2/8; the diagonal is exactly twice the background in paper coverage in the limit, and the anti-diagonal is its pixel-for-pixel copy by the palindrome symmetry. Witness: moire-correlation-laws.
• [Proved] The moire rays obey a 2-adic law, not a Farey law: the slope-one family at offset a/b carries (-1)^a/b^2 for odd b and exactly nothing for even b, and the slope q/p ray through the origin carries exactly 1/(pq), the same number as the correlation of grams p and q. Witness: moire-correlation-laws.
• [Verified] The stack fades at the random rate in L^2: RMS contrast falls as c/sqrt(L) with c^2 = lim L * Var, c = 0.522, a constant factor 1.2054 above independent layers, so "does not fade like random noise" is false in L^2; the exact variance is a finite rational sum at every L. Witness: moire-correlation-laws.
• [Proved] Coprime independence holds exactly in all four flat families: carpet, net, tree and void layer pairs have covariance identically 0 at every coprime odd pair, checked exhaustively to 201 and in exact rationals at (3,5), (5,7). Witness: moire-correlation-laws.
• [Proved] The void flat stack obeys a gcd-to-the-fourth law, Pearson_void(m,n) = (g^4 - 1)/sqrt((m^4 - 1)(n^4 - 1)) (void being the pure pair field (1 + s(mu) s(mv))/2), tree obeys (g^2 - 1)/sqrt((m^2 - 1)(n^2 - 1)), net's closed form is carpet's under m, n -> -m, -n with slightly larger correlations, and the gcd echo orders tree > net > carpet > void; all four match exact lcm-grid counting on (3,9), (5,15), (9,15) with zero error. Witness: moire-correlation-laws.
• [Verified] The flat variance constants reduce to two gcd sums, S2(N) = sum g^2/(mn) and S4(N) = sum g^4/(m^2 n^2) over odd pairs: lim L * Var is the S2/(2N) limit for tree, the S4/(2N) limit for void (0.2768062) and S2/(4N) + S4/(8N) for carpet and net (their difference dying like ln^2 N / N), with the identity Var_L(carpet) = Var_L(tree)/2 + Var_L(void)/4, measured to 5 digits. Witness: moire-correlation-laws.
• [Proved] The S4 limit is a theorem: lim S4(N)/N = (16/31) T/zeta(5) with T = sum_{k,l odd} 1/(k^2 l^2 max(k,l)) = 1.1122336970, value 0.5536124372, measured 0.5536124482 at N = 3 * 10^6, by a bounded coprime tail plus Mobius over odd moduli, with an independent Jordan-totient recomputation. Witness: moire-correlation-laws.

Diagonal slice stack

• [Verified] The level-1 slice is exactly a lattice-plane object: carpet_cut(n, 1) equals the set x + y + z = 6n - 2, z even, in [0, 4n)^3, filled iff at most one of floor(x/4), floor(y/4), floor(z/4) is odd, with |slice| = 6n^2; two cell-for-cell reconstructions (n = 1..63 and n = 1..13) show zero mismatches. Witness: walsh-spectrometer.
• [Proved] The 1:6:1 three-plane law: every micro point of the slice lies in a macro cell with i + j + l in {K, K-1, K-2} at K = (3n - 1)/2 with multiplicities 1, 6, 1, so fill(n) = 6 F(K-1) + 2 F(K), a two-line derivation of the cut fill closed forms, exact for all odd n <= 21 and holding at n = 1 where the outer planes are empty. Witness: walsh-spectrometer.
• [Proved] The slice's two mod-4 families are the Dirichlet character chi_4: per-gram ink is exactly 3/8 + 1/(2n) + 1/(8n^2) at n = 1 mod 4 and 5/8 + 1/(2n) - 1/(8n^2) at n = 3 mod 4, because the plane constraint pins the triple parity product to (-1)^K; 14 + 14 layers cancel it, so the stacked snowflake sits at background 1/2 while the flat carpet stack sits at 3/4 ink; the closed forms reproduce all 28 layers with zero error. Witness: walsh-spectrometer.
• [Verified] The chi_4 twist kills the pair-resonance ray family: the average of chi_4(n) T(nx) over odd n <= N falls like 1/N at every x, rational or not (x = 0, 1/3, 2/3, 1/5, 1/7, 1/2, 1/4, 1/9 and irrational), so the snowflake stack has no analogue of the carpet stack's bright main diagonal, its A = C excess going -0.0036 at N = 55 to -0.000052 at N = 5555; its visible rays are only the three single-wave crosshair families parallel to the hexagon's edge directions, one per lattice axis, at odd-denominator rational coordinates. Witness: lab/hexagon-moire.
• [Verified] The ghost star at the hexagon's centre is a finite-layer artifact: each layer's centre is entirely ink or entirely paper, flipping with n mod 4, so 28 layers give exactly 1/2, which is also the limiting background; the star-minus-background contrast decays as (ln L)/L (-0.094 at 5 layers, -0.031 at 28, -0.018 at 56; excess * L running -0.7779 to -1.2519 in the ideal frame and -1.0212 to -1.3645 in the lattice frame from L = 28 to L = 400), and the exact rate constant is open and frame-dependent (-1/8 per ln L in one frame, -0.18 in another). Witness: lab/hexagon-moire. Superseded in the cell frame: the decay coefficient is a closed form at every band width, see the width family rows under Diagonal slice stack in SETTLED.
• [Verified] The three 60-degree crosshair families obey a limit law: the line at coordinate a/q carries strength 1/(4q) for odd q and nothing for even q, converging in the arithmetic model (1/3 -> -0.0837 against -0.0833 at N = 5555) and visible in the real render in registration-correct frames (the X + Z = 1.25 line at N = 55: -0.045/+0.029; the X = 1/3 one-sided bands +0.021/-0.058); the per-layer registration drift of 1/(2n) in the slice plane is what a drifted scan raster misreads (1/7 at -0.058 against -0.036, 1/2 at -0.014 in the coarse crosshair model), and a null claiming no rays above 0.013 was about the scan geometry, not the object. Witness: lab/hexagon-moire.
• [Proved] The Walsh spectrometer: the diagonal-slice ink of every 3D parity design is the exact quasipolynomial ink_D(n) = Sig0 - (1/2) Sig3 s + [(2/3) Sig1 - (1/3) Sig2 s]/n + [(2/3) Sig2 - ((1/3) Sig1 + (1/2) Sig3) s]/n^2 with s = (-1)^((3n-1)/2) and Sig_j the design's level-j Walsh coefficient sums: the background is the mean Walsh coefficient, the mod-4 blink is minus half the top coefficient, the 1/n orders read the middle levels; exact in rationals on all 256 codes at every odd n <= 55 and at the cold sizes 101, 555, 999, 9991; the attempt to break it recomputed P_n from the definitions for all 28 odd n <= 55 independently of the lane's scripts and of the crate, found |P_n| = 6n^2, weight-only dependence with zero splits and zero law mismatches on all 256 codes. Witness: walsh-spectrometer, mrlyweb::walsh_spectrum.
• [Proved] Nine of the 22 design classes never blink: |b| takes exactly the values {0, 1/16, 1/8, 3/16, 1/4}, zero iff the top Walsh coefficient vanishes (tree and void among them), carpet and net blink at the middle rung 1/8, the xor pair maximally at 1/4; (a, |b|) is orbit-invariant on all 256 codes, the named codes are carpet 23, net 232, tree 3, void 129, and carpet and net are the same symmetry class (net is carpet with all parities flipped). Witness: walsh-spectrometer.
• [Verified] The xor pair blinks hardest: code 105 has ink(n) = 1/2 - s/4 - s/(4n^2) and its complement code 150 the reflection 1/2 + s/4 + s/(4n^2), both swinging 1/4 to 3/4; the attempt to break it checked n = 1, where 150 inks 0 of 6 cells against the 1 the shared formula would demand. Witness: walsh-spectrometer.
• [Verified] The corrected law on the hexagon is dyadic: what breaks coprime independence is a hidden half-cell-shifted overtone at doubled frequency, chi_4(n) s(2nX + 1/2)/8, that the plane constraint forces into every carpet slice, plus the hexagon's non-product tent marginal; together they couple layer m to layers 2m +- 1 and m +- 2 regardless of gcd, and the doubling sign law sign r(m, 2m +- 1) = -chi_4(m) chi_4(2m +- 1) holds on 18 of 18 pairs from (3, 5) to (601, 1201) across all four residue branches. Witness: lab/hexagon-moire.
• [Verified] The doubling magnitude reads between 0.11711630 and 0.11715991 (Richardson extrapolation on sliding triples of m = 157..601), and the two branch extrapolations in 1/m land on 0.1171270 and 0.1171274, so the exact rational -19/162 = -0.117284 is dead at 1.57e-4. Witness: lab/hexagon-moire.
• [Verified] The breakage is rule-specific in the limit: the persistent doubling coupling is carpet and net only (void and tree doubling correlations die, +0.001 at (201, 401)), tree keeps a neighbour coupling r(m, m+2) -> -0.0704, void stays essentially independent (adjacent +0.008), and the gcd echo survives in all four (carpet (m, 3m) -> +0.2148, tree +0.1498, void +0.0772 at (67, 201)); on the full hexagon the coprime pairs (5, 9) and (5, 7) read -0.142 and -0.085, so any published number must pin the mask convention. Witness: lab/hexagon-moire.
• [Verified] Eisenstein is absent from the base-2 slice stack: L(2, chi_-3) = 0.7813024129 appears nowhere, the only character the slice generates is chi_4, and the hexagonal geometry contributes rational tent integrals. Witness: lab/hexagon-moire. Superseded on the character claim: the arm of the ghost star carries chi_8, see the width family rows under Diagonal slice stack in SETTLED.
• [Verified] The quarter-line law: the strongest interior lines of the stacked hexagram sit at quarter-cell coordinates a/4 (generally a/(4b), b odd), an exact one-sided step of +-1/8 that every layer votes for identically because the overtone's chi_4 sign meets the layer's own chi_4 and squares away, converging 0.1221, 0.1234, 0.1241, 0.1245 at N = 151, 301, 601, 1201, the odd-fraction crosshairs at 1/(4q) following behind; the X, Z, W profiles are numerically identical on the render by the slice's permutation symmetry, so "in five directions but never horizontal" is false and the missing-Z-overtone statement holds only in the rectangle-cell frame. Witness: lab/hexagon-moire.
• [Verified] The void slice stack keeps its star forever: six central lines at plateau ink 1/2 against background 1/4 (ratio 2, the model-frame Z arm weaker at 3/8), every layer voting on all six, plus a centre dot that is ink at every odd n by a two-line parity proof; the carpet's star fades as log-corrected 1/L, so the two snowflake stacks differ by a theorem. Witness: lab/hexagon-moire.
• [Verified] Void and carpet have complementary line spectra on the cut: void's lines sit at even-denominator twisted positions X = a/(2b), b odd, where carpet is silent, and void is silent at carpet's odd rationals; tree carries the only untwisted crosshair family plus a permanent ratio-2 line at K = 3/2 and ignores its free axis; net is the exact pixelwise complement of carpet, since "at most one odd" and "at least two odd" exhaust the cases. Witness: lab/hexagon-moire.
• [Proved] The cut ink laws of all four families are exact closed forms with chi = (-1)^((3n-1)/2): carpet 1/2 + chi/8 + 1/(2n) - chi/(8n^2), net 1 - carpet, tree 1/4 + (1/3 - chi/12)/n + (1 - chi)/(6n^2), void 1/4 - chi/(4n) + 1/(2n^2), exact for all odd n <= 55 in lab/hexagon-moire; the wider range n <= 101 has no generator. Witness: lab/hexagon-moire, walsh-spectrometer.
• [Proved] The slice stacks' surviving constants are Leibniz, odd Basel and Catalan: the carpet split M (I1 - I3 + 1/4) -> pi/4 + pi^2/32 unconditionally at balanced layer counts (1.08267 at M = 28 against 1.09382), the void background (pi + pi^2)/16 = 0.8131998159 (a fourth-decimal near-collision with the flat-stack pi^2 ln 2/(7 zeta(3)) = 0.8130217042, explicitly separated), the tree background pi/48 + pi^2/48 + G/6 with G Catalan's constant, all character series over the exact ink laws, to 7 digits each. Witness: lab/hexagon-moire.
• [Proved] The ghost star's decay coefficient is a closed form at every band half-width, not just at the arm. Widen the star to the band |x - y| <= W cells; x - y is even on the cut, so W enters only through K = floor(W/2). With b = 1 when floor(K/2) is even, chi = (-1)^((3n-1)/2) the ink law's character (-1 at n = 1 mod 4), chi_8 the real character mod 8 of Q(sqrt 2), and E(K) = #{|j| <= K : j = 3, 4, 5 mod 8} + floor((K + 2)/4) - K the block tail's chi_8 weight, the band's excess over the hexagon's ink law is exactly kappa chi + (m + q chi_8(n))/n + chi/(8 n^2) at every odd n >= K, with kappa = -(-1)^K/(8(2K + 1)), m = -(K + b)/(2(2K + 1)) and q = (1 - 2E(K))/(2(2K + 1)); below n = K the band is clipped and the identity is false, W = 6 at n = 1 missing by 2/7. The decay coefficient is therefore -(K + b)/(4(2K + 1)), the conjectured -(W + 2b)/(8(W + 1)) at even W and -(W - 1 + 2b)/(8W) at odd W, tending to -1/8. E is 8-periodic because a block of eight adds 6 + 2 - 8 = 0, matching the run 0, -1, -1, 0, 1, 2, 2, 1 at 201 of 201 values K = 0..200, and the identity matches the counted band in exact rationals 1354 of 1354 at 14 distinct half-widths, every odd n from K to 201, with the four classes n = 1, 3, 5, 7 mod 8 counted apart. Witness: lab/hexagon-moire.
• [Proved] The width family's constant and both its 1/L^2 branches are closed forms at every width. At even L the ladder is L * excess_L = (m/2) ln L + C_W + O(1/L^2) with C_W = m (ln 2 + gamma/2) + q L(1, chi_8) - G/8 + Delta_W, where Delta_W is the sum over odd n < K of the counted excess less the identity, the exact rational the clipped layers contribute, 0 through W = 5 and 2/7 at W = 6. The chi_4 components cancel at the 1/n order only, so L(1, chi_4) = pi/4 is absent at every width while L(2, chi_4) = G sits in every one: C_W carries gamma, ln 2, L(1, chi_8) and Catalan's G. Three tails give the 1/L^2 coefficient: the chi_8 tail over odd n > 2L is -q/4 at L = 0 mod 4 and +q/4 at L = 2 mod 4, since the sign pattern +--+ on the four odd residues starts at n = 2L + 1; the Catalan tail of the background's chi/(8 n^2) gives +1/64 blind to the residue; and the harmonic remainder of m (H_{2L} - H_L/2) gives +m/48. So the coefficient is -q/4 + 1/64 + m/48 against +q/4 + 1/64 + m/48, which at W = 0 is -23/192 and +25/192. The sliding-window slope the sweep reads cancels the oscillation only at L = 0 mod 4 and converges to m/2 + kappa/ln 2 at L = 2 mod 4: the generator reads -0.24999980 at L = 1600 against m/2 = -1/4, and -0.43078703 at L = 1602 against the limit -0.43033688. Witness: lab/hexagon-moire.
• [Proved] For a fixed affine phase map n = a m + c with m growing inside one class mod 4, and on the mask this page always uses - the full hexagon of the common cut, area-weighted exactly - the layer-pair correlation limit is an exact rational; a general pair (m, n) has no limit theorem here. The cut cell obeys s_y = s_x + s_z + w mod 2 with w = 1 at exactly the phase cells (p, q) = (0, 0) and (3, 1) when N = 1 mod 4 and its complement when N = 3 mod 4; the map sends alpha = mX mod 2 to (a alpha + c X) mod 2, and (mX mod 2, mZ mod 2) equidistributes on the fixed polygon at O(1/m), leaving a piecewise-constant integral with rational breakpoints. It returns the doubling constant exactly 253/2160, covariance 253/9216 over variance 15/64, at all four branches with the sign law's sign; the adjacent limit exactly -11/135 and the gcd echo exactly 29/135; the tree 0, -61/864, 4/27 and the void 0, +7/864, 2/27, the two doubling zeros exact. 19/162 is refuted. Both residue classes converge: (301, 601) reads -0.11745304 and (601, 1201) -0.11729091 at m = 1 mod 4, (103, 205) reads +0.11914004 and (203, 405) +0.11814528 at m = 3 mod 4, gap times m at -0.097, -0.097, +0.207 and +0.206. Witness: lab/hexagon-moire.
• [Proved] Every constant of the stack's recentred one-layer cut ink is one character sum, and the layer count's parity is the only residue it reads. The hypothesis carries two limbs: the object is the cut ink of a 3D parity design, whose Walsh quasipolynomial carries only chi_4 and terminates at the 1/n^2 order by the Walsh spectrometer's ink theorem, and the quantity is the recentred M (mean - A - c eps). Writing a family's ink law as I(n) = A + B chi + (c + d chi)/n + (e + f chi)/n^2 with chi = -chi_4(n), the average over the first M odd sides obeys M (mean - A - c eps) = -B S - d s_1 + e s_3 - f s_2 exactly, with eps the mean of 1/n, three chi_4 sums and the zeta tail s_3 = sum 1/n^2 over those layers, so the limit is -B [M odd] - d pi/4 + e pi^2/8 - f G: pi enters only through the 1/n order of the ink law, Catalan only through the 1/n^2 order, the residue class only through B, and inside those two limbs no other constant can appear, so L(2, chi_-3) is absent by a theorem rather than by a search; outside them it is not, the ghost star's width family being a one-layer object of the same stack whose constant carries gamma, ln 2 and L(1, chi_8) because its character is mod 8 and its 1/n limb is not subtracted. The four families read (A, B, c, d, e, f) as (1/2, 1/8, 1/2, 0, 0, -1/8), (1/2, -1/8, -1/2, 0, 0, 1/8), (1/4, 0, 1/3, -1/12, 1/6, -1/6) and (1/4, 0, 0, -1/4, 1/2, 0), every row of the constants table is an instance, and the leading term A + B chi is the pair sections' own phase-map integral taken at the identity map a = 1, c = 0. The generator holds the summed identity against the counted hexagons in exact rational arithmetic at every layer count to N = 55, all four families, the classes n = 1, 3, 5, 7 mod 8 counted apart, 7 of 7 in each. Witness: lab/hexagon-moire.
• [Proved] The Catalan statement holds at both residue classes and neither one is a fit. The carpet is B = 1/8, d = e = 0, f = -1/8, so M (mean ink - 1/2 - eps/2) -> G/8 = 0.1144956993 along N = 3 mod 4, measured 0.1144757884 at N = 55, and -> G/8 - 1/8 = -0.0105043007 along N = 1 mod 4, measured -0.0104828892 at N = 53. The 1/8 step is the ink law's own chi averaged over an odd number of layers, the same parity term the ghost star's even-L hypothesis carries, and Catalan enters only as L(2, chi_4), one order below the pi the tree and the void collect. This closes the Conjecture of the same name. Witness: lab/hexagon-moire.
• [Proved] The approach to every one-layer constant is a closed form. With sigma = +1 at even M and -1 at odd M, the three tails past a = 2M + 1 solve T(a) + T(a + 2) = a^-s twisted and T(a) - T(a + 2) = a^-s untwisted in powers of 1/a, giving sigma/(4M), sigma/(8M^2) and 1/(4M) with the 1/M^2 limb of each cancelling, so the gap to the limit is (sigma d - e)/(4M) + sigma f/(8 M^2) + O(1/M^3). Carpet and net read gap times M^2 as -1/64 at even M and +1/64 at odd, the void gap times M as -3/16 and -1/16, the tree as -1/16 - 1/(48M) and -1/48 + 1/(48M), and the carpet split as -5/16 at even M only. The generator's ladders at M = 400, 1600, 3200 print all four classes of M mod 4, so all four of N mod 8, and match to eight decimals at M = 3200. Witness: lab/hexagon-moire.

Dimension one

• [Proved] The pincer at dimension one: Lemma G, the gasket case of Lemma B, hence all 36 lines by the reduction above, holds for every prime exponent beta = log_3(p)/n below 0.4475978 and above 0.6402122: below by exact gasket moment identities (carry-free additive energy exactly 15^a, 6th, 8th and 10th moment growths the exact algebraic numbers 57 + 6 sqrt(46), 456 + 3 sqrt(11017) and the largest root of x^4 - 7833x^3 + 7916949x^2 - 850684437x + 13054946580 from nine- and twenty-five-state transfer matrices) fed through a Holder ladder that never uses ord_p(3), above by the ray decomposition (fibres 2^(n+1) - 2, shift rays at most 2n phi^n, every carry state of every primitive ray at most 2 admissible digits) with the regime bookkeeping on the trichotomy of n against 3a and 4a, a = floor(log_3(p/2)), which closes the belt of primes near p ~ 3^(n/3); the ladder saturates at 2/(3 + log_3 5) = 0.447931 < 1/2 and per-ray-maximum methods stop at 1/2; the master bounds hold against exact L_n(p) for every prime 5 <= p <= 199; the eighth rung 0.446717 is the row the shelf lane still imports and is now one row stale. Witness: lemma-b-pincer, lab/dimension-one-ladder.
• [Proved] The burst certificate moves the pincer's top edge to 0.6402122 unconditionally: when 3^k divides a ray coordinate (div) or the coordinate sum (opp), the branching type of a carry state is predetermined k steps ahead and the two digits of a 2-branching state force distinct types exactly k steps downstream, so admissible paths inject into the subsets of {1..w} avoiding distance exactly k, a Fibonacci product bounded by phi^(w+k) and attained on the shifts (9,1) and (27,1), hence rho <= phi for every primitive ray, one certificate for the infinite family with no computation; the opp designation rule at states whose predetermined window hits the dead type is fixed by a digit-order tie rule never consulted in the argument, the div ray count is per orientation with constant 61.6 rather than 600, and the edge rounding 0.640212 sits 1.9e-7 on the unsafe side. Witness: lemma-b-pincer.
• [Proved] Theorem R: the second moment Z(n) = sum_y M_n(y)^2 over primitive rays satisfying Z <= C 3^(gamma n) closes every beta > gamma/2, with 1/2 the method's own wall since the diagonal alone forces Z >= 3^n; Z(n) decomposes exactly as diagonal plus multiplier triples (s, t, z) with Q_n(1, 3^j) in closed form, Z/3^n peaks at 2.676455 at n = 10 and falls monotonically to 2.226210 at n = 18 (limit 2 conjectured, which would shrink the window to (0.4475978, 1/2]); directions are reduced from the origin per point, never as pairwise displacements, the six-digit x_j + y_j <= 2 construction is a different 6^n gasket and not G_n, and the 1D coprime pairs of the {0,1} base-3 Cantor set (Z/4^n -> 0.513358) are a different object. Witness: lemma-b-pincer.
• [Proved] Two named ways past 0.6402122 are shut: keeping the exact Fibonacci burst product D_k(w) = prod_r F_(m_r + 2) cannot lower the octave exponent, since at k = 1 already D_1(w) = F_(w+2) = Theta(phi^w) while the depth-one octave-j census is at most a constant times 3^(2j-1), reproducing psi_phi(c) = 2c + (1 - c) log_3 phi and beta = 1/(2 - log_3 phi); and averaging spectral radii cannot replace the supremum, since ray mass is governed by rho^w and Jensen gives int rho^w dnu >= (int rho dnu)^w; the height-40 catalogue (486 primitive non-shift rays plus the shifts (1,3), (1,9), (1,27)) has mean rho = 1.0228285, median 1, standard deviation 0.0907378, inverse-square-weighted mean 1.0639086 (a 34.25% deficit below phi, 1.0481989 if (1,1) were wrongly counted as a shift) and certified weighted upper mean 1.0997454, and substituting them into (2 - log_3 lambda)^(-1) yields 0.5145062, which is not a coprimality bound and must never be quoted as one; the inverse-square weight is a probability measure only after a height cutoff since sum 1/(a^2 + b^2) diverges logarithmically, and the shifts are removable because their count is O(1) per octave. Witness: lemma-b-pincer.
• [Verified] Paley-Zygmund, Bonferroni and Cauchy-Schwarz on the first two Fourier moments are structurally unavailable for the dimension-one lower edge: Paley-Zygmund lower-bounds the heavy rays while the proof needs an upper bound on the total bad mass sum_y M_n(y); Bonferroni needs the signed intersection counts T*_pq, T*_pqr, ... with no uniform estimate over the exponentially growing modulus range; and (sum_t F_a(t))^2 <= p^2 sum_t F_a(t)^2 is an upper bound on the first absolute moment, the direction the ladder already uses, so M_1^2/M_2 cannot improve 0.4479; the cap 0.447930987882 is purely the absolute-Fourier-moment wall from the low-frequency peak E_2K >= 3^((2K-2)a)/K^2, forcing kappa_2K < 2 at every finite K, not a Cauchy-Schwarz artifact and not Mobius truncation (which needs the separate tail control A_z(n) - A(n) <= G(n)/log z, divergent at dimension one); no universal cap holds for "any moment-based method", since a complete moment sequence determines the distribution. Witness: lemma-b-pincer.
• [Verified] Higher Fourier moments cost polynomial time in the moment order: once the carry transfer matrix is built, E_2K(G_a) = (M_K^a)_{0,0}, polynomial in the level a by matrix powering; the bounded carry radius is about K/2, giving S_K = (2 c_K + 1)^2 = O(K^2) states, and the naive construction is about O(K^6) operations before bit complexity, with 9, 25, 25, 49, 49, 81 states at orders 6, 8, 10, 12, 14, 16; exact characteristic-polynomial algebra still grows with integer size, and a numerical Perron root is not a master inequality. Witness: lab/dimension-one-ladder.
• [Verified] The multiplier pairs have a spectral gap at 2: lambda(s,t) = 3 only on the shift pairs (1, 3^j) and their reverses, every other coprime pair obeys P_w <= (3/2)^K 2^w at every state with K = v_3(st) + v_3(t' - s'), and the interval (2, 3) is empty over the certified domain max(s,t) <= 52 only, universality being the lane's open con:gap and no theorem; aligned 3-way splits land their penalty exactly k steps later, giving G_m = G_(m-1) + 2 G_(m-2) with Perron root exactly 2 at (1,4) (P_w = (2^(w+2) + (-1)^(w+1))/3, characteristic polynomial lambda (lambda - 2)(lambda + 1)); over all 829 coprime unordered pairs with max(s,t) <= 52, 20 non-shift pairs attain 2 and the largest non-shift radius below 2 is 1.6956207695598 for the gasket-digit automaton A but 1.8488475886485 for the free-digit B the census actually needs, on (4,13), (4,39), (12,13), (13,36); the 9-divisible classes (9,2), (2,9), (18,1), (1,18) have lambda = 1; the closest ratios to the bound are 0.8888893 at (3,4), (3,7), (1,12) and 0.8888887 at (1,4), (1,7); the gap means no radius strictly between 2 and 3, not a gap below 2, and alone gives only E <= C 9^n. Witness: gasket-ray-machine.
• [Verified] The heavy gasket rays carry named run-length counts: M_n(3,1) = F(n+1) - 1, the Fibonacci product prod_r F(m_r + 2) - 1 at (3^j, 1), Narayana's cows A000930(n) - 1 at the supergolden ray (1, 12), and c(n-3) - 1 with c(m) = c(m-1) + c(m-4) at (7, 3), exact to n = 140 (M_140(3,1) = 131151201344081895336534324865, M_140(1,12) = 106502839316458556100416, M_140(7,3) = 21561294536157802712); M_14(7,3) = 49 = 7^2 is a coincidence, x^4 = x^3 + 1 being irreducible and the quartic sequence square only at n = 4, 7, 9, 12, 14 (1, 4, 9, 25, 49) through n = 140; max M_13 = 376 = F(14) - 1. Witness: gasket-ray-machine, A000930.
• [Proved] Codes 98, 140, 266 and the fourth permutation design {(0,2),(1,1),(2,0)} (code 84 under the 3a + b indexing) are diagonal: Z_F(n) = 3^n - 2 for every n >= 1 (the identity fails at n = 0, where the sides are 0 and -1), no two distinct points ever collinear with the origin, by a 3-adic cross lemma: weights all in one unit residue class mod 3 against weights injective mod 3 pin the cross determinant's valuation to the first differing digit position; the three named codes are the permutation graphs j -> j+1, j -> j+2 and the swap of 0 and 1, so the lemma has instances and not separate proofs; for these designs ray mass is ray occupancy and the window problem is pure divisor rarity. Witness: gasket-ray-machine.

Divisor avatars

• [Proved] The Avatar Theorem: for x = prod p_i^(a_i) with p_1 < ... < p_D and every a_i >= 1, the dimension-D design with f_w = e_w(a_1 - 1, ..., a_D - 1) has P(n) = d(x^n) at every n >= 0, by the substitution a_i n + 1 = b_i n + (n + 1) and prod (b_i t + 1) = sum_w e_w(b) t^w at t = n/(n+1); the map is injective by Newton's identities and surjective onto the f_0 = 1 signatures whose fill splits completely into linear factors over Q, and a design exists iff sum_i (a_i - 1) <= D; the first ten colossally abundant numbers 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720 have avatars while the first without one is 21621600; on the seven D = 3 ladders x = 30, 60, 120, 180, 240, 360, 900 to n = 20, all 131 of the 140 powers exceeding 5040 satisfy Robin's inequality, the largest ratio R(14400) = 1.5732599059 against e^gamma = 1.7810724180, margin 0.2078125121, strictly decreasing in n on every ladder, which proves nothing about Robin beyond them; sigma(m)/m < sigma(N)/N for every m < N defines a superabundant number, not a highly abundant one. Witness: divisor-avatars.
• [Proved] The sigma-hunt is closed negatively for polynomial census laws: no census law of a design at odd side 2n + 1 equals sigma_k(x^n) for k >= 1, x > 1, since that grows at least like x^(kn) while every such law is a polynomial in n of degree at most D, and none equals sigma(x^n)/x^n, strictly increasing and bounded hence not constant; at x = 1 both collapse to the nine constant identities O(n) = 1 = d(1^n); over the nine observables (fill, voids, surface, touched vertices, edges, faces, Euler characteristic, components, cycle rank) on all 22 least-mask representatives of the 256 base-2 3D designs counted through side 21, exactly eight non-fill strict divisor avatars survive, only for voids, Euler characteristic and components; E and R vanish at n = 0 while d(x^0) = 1, so they are never strict identities, and the four graph laws for codes 30 and 126 hold only for n >= 1; if sigma is a design observable at all it lives among geometrically growing counts. Witness: divisor-avatars.
• [Verified] The eight non-fill divisor avatars come from topology, not measure: the empty design's voids (2n+1)^3 = d(900^n); Euler characteristic and components both (n+1)^3 = d(30^n) for {000}; both (n+1)^2 = d(6^n) for {000, 100}; components n + 1 = d(2^n) for the three-corner path {000, 100, 010}; both n + 1 = d(2^n) for the square face {000, 100, 010, 110}; plus nine constant identities O(n) = 1 = d(1^n), components for codes 23, 27, 31, 61, 63, 111, 127, 255 and Euler for 255; the statement is for the canonical least-mask representatives, since cube symmetry does not preserve the odd/even origin under coordinate reversal; codes 15 and 27 share a weight signature and a fill law, so eight orbits carry seven weight signatures. Witness: divisor-avatars.

Kronecker words

• [Proved] Fill, side, density and the main-diagonal count of a mixed Kronecker word are order-blind at every word length, the first three as products of per-factor quantities and the diagonal by diag(A (x) B) = diag(A) (x) diag(B), exhaustive on all 15^3 words of length 3 at base 2, D = 2; the whole anti-diagonal profile is order-sensitive on 99 of 105 multisets at length 2 over the 15 non-empty codes and on 204 of 210 at length 3 over the ten-code library of every code of fill 2 or 3, its peak on 23 of 105 and its support on 27 of 105, minimal witness the one-cell codes 1 and 2 with profiles (0,1,0,0,0,0,0) against (0,0,1,0,0,0,0); take the whole profile, never one coefficient. An earlier length-3 reading of 110 of 112 is withdrawn, not confirmed: 112 is the multiset count of an eight-code library that was never recorded, so the figure names no sweep anyone can rerun. Witness: lab/magic-words.

Slice sign law in every dimension

• [Verified] The slice sign law holds on every computed range: the central diagonal slice of the D-dimensional base-3 Menger analog has sgn(dim_slice - (d - 1)) = (-1)^(D+1) for D = 2..50 at 180 to 210 digits by two generators sharing no code, for D = 2..100 at 320 digits with 99 of 99 signs, and in exact rational arithmetic through the determinant form sgn det((fill/3) I - M_even) = (-1)^D for D = 2..40, with fill = 2^(D-1)(D+2) and M_even the reflection-even carry block of size ceil(D/2); base 5 alternates for D = 2..15, all four tested non-Menger families alternate, off-centre heights keep the dominant eigenvalue; D = 3 gives x^2 - 9x + 12 with rho_3 = (9 + sqrt(33))/2 = 7.372281 against fill/3 = 20/3, D = 4 gives x^2 - 11x - 66 with rho_4 = 15.310708 against 16 and det = 14; the even half and the odd D != 1 mod 3 half are proved on the shelf, and the class D = 1 mod 3 beyond the computed range stays open. Witness: slice-sign-even-half, slice-recurrence-order.
• [Verified] The D = 3 rung is the base-3 slice dimension: the carry automaton M[c, c'] = P[c + D - 3c'] prints M_even = [[6, 6], [1, 3]], trace 9, determinant 18 - 6 = 12, characteristic polynomial x^2 - 9x + 12, exactly A299916's signature (9, -12), Perron root (9 + sqrt(33))/2 and log_3 of it 1.818410, against d - 1 = log_3(20) - 1 = 1.726833 with fill = 20 the sponge's surviving-subcube count; the anchor cuts one way only, saying nothing about higher rungs. Witness: slice-recurrence-order, A299916.
• [Proved] The digit polynomial P(t) = (1 + t^2)^(D-1) (1 + D t + t^2) has B_D(2k) = C(D, k) and B_D(2k+1) = D C(D-1, k), P(1) = 2^(D-1)(D+2), P(-1) = 2^(D-1)(2 - D), P(omega) = (-1)^(D-1) (D-1) omega^D, and root-of-unity filtering gives the full carry matrix's exact row sums sigma(c) = fill/3 + (2/3)(-1)^(D-1)(D-1) cos(2 pi c/3); the row-sum identity holds in the carry orientation c -> (c + D - s)/3 and fails in the transposed even-basis orientation M_even[i,j] = B_D(D + j - 3i) + B_D(D - j - 3i) for every D = 3..50, the D = 3 row sums being (12, 4) against the formula's (8, 6); the coefficient formulas hold at D = 1..10 three positions past both polynomial endpoints. Witness: slice-recurrence-order.
• [Proved] The trace of the even carry block is tr(M_even) = 3 D 2^(D-3) at odd D and 3 * 2^(D-2) - 1 at even D, reading 2, 9, 11, 60, 47, 336 at D = 2..7; the even case's -1 is real, starting at D = 2 where the matrix is [2]. Witness: slice-recurrence-order.
• [Verified] There is no uniform spectral gap in the slice transfer matrix, so no fixed-epsilon proof of spectral separation can exist: lambda_1/|lambda_2| = (D+2)/(D-2) + O(D^-3), tending to 1, reaching 1.068966 at D = 60 and 1.04081632653 at D = 100, 1.0833... at D = 50 against 13/12 to 2.58e-22, with lambda_1 ~ fill/3 = 2^(D-1)(D+2)/3 and |lambda_2| ~ |P(-1)|/3 = 2^(D-1)(D-2)/3; the double-precision spectrum agrees with a 180-digit reference over D = 2..50 to worst relative Perron discrepancy 2.3e-15, median 4.7e-16, every eigenvalue numerically real over D = 2..60, so any proof of separation must be uniform in a margin of order 4/D. Witness: slice-recurrence-order.

Diagonal slice ladder

• [Verified] The even half at base 7 carries exact Collatz-Wielandt certificates rho_D < fill/7 at every even D <= 26 and at D = 172, 174, anchored by P(1) = 6^(D-1)(D+6) and a brute-force digit enumeration of P, with depth K_min = 0, 1, 2 stepping at D = 4 and D = 26 (and 3 at D = 174), V(L) > 0 at every even D = 2..40, L <= 25 with no dip, and c == 0 mod 7 immune at depth 0, so Sigma_K depends on c mod 7^(K+1). Witness: slice-sign-even-half.
• [Verified] An off-centre diagonal slice cannot break the alternation: a fixed target offset k changes only the initial vector, the transfer matrix commutes with carry reflection and e_0 is even, so e_k^T M^L e_0 = (1/2)(e_k + e_(-k))^T M^L e_0 and the odd component is annihilated, the off-centre slice seeing only the central even block; the boundary condition matches direct polynomial multiplication in 24 cases at D = 2..7, L = 1..4, exact counts to L = 80 obey minimal rational recurrences on every term, and the dominant root equals the central one at every D = 2..12 and offset |k| <= 2, the only movement being transient zero modes at D = 4, |k| = 2 and D = 2, |k| = 1, 2 that raise the order without touching the growth rate; offsets scaling with 3^L are untested. Witness: slice-recurrence-order.
• [Verified] The anti-diagonal slice profile factors across the Kronecker product, P_(A (x) B)(t) = P_A(t^(n_B)) P_B(t), since r + c = n_B (r_A + c_A) + (r_B + c_B), giving the stationary product prod_(j<L) P(t^(q^j)) and its mixed-radix form, so one identity covers fractal slices, mixed-product slices and the dimensional ladder; exact on all words of length 2 and 3 at base 2, D = 2, zero mismatches. Witness: lab/slice-ladder-controls.
• [Verified] The level-1 central diagonal slice of the base-3 Menger analog is the vertex set of the cube's central cross-section: the hypersimplex vertex counts 2, 6, 6, 30, 20, 140, 70 at D = 2..8 match C(D, D/2) for even D and C(D, (D-1)/2)(D+1)/2 for odd, so the ladder starts on a polytope rather than on an analogy. Witness: lab/slice-ladder-controls.

Hexagram provenance

• [Verified] The hexagon-triangle substitution matrix [[6,1],[6,3]] (an encoding of published prose, "replace each hexagon with 6 hexagons and 6 triangles, and replace each triangle with 1 hexagon and 3 triangles", not a published matrix) iterated from (1,0) gives 1, 6, 42, 306, 2250, 16578, the A299916 row term for term, by matrix iteration and by the (9,-12) recurrence alike; one side counts tiles and the other holes, so the shared row is not by itself an identity of objects. Witness: A299916; slice-recurrence-order.

Stacked hexagon moire

• [Verified] The Walsh law's 1/n and 1/n^2 coefficients follow from plane counting: the plane x + y + z = 6n - 2, z even, 0 <= x,y,z < 4n splits by macro parity into counts that depend only on the weight k, N_k(2h+1) = [t^(3h+1)](1 + 6t + t^2) E_h(t)^(3-k) O_h(t)^k with E_h = 1 + t^2 + ... + t^(2h) and O_h = t + t^3 + ... + t^(2h-1) extracts all four exact quadratic quasipolynomials by finite binomial calculation, and ink is linear in the eight counts, so only the constant, 1/n and 1/n^2 orders survive the normalization; exact by direct enumeration of the plane at every odd n <= 55, all eight macro-parity triples, 28 layers, zero discrepancies; the crate rebuilds only the ink, at codes 23 and 11 over n = 1..11, where m_3 = 0 leaves N_3(n) unexercised, so the eight-triple split stays the lane's. Witness: walsh-spectrometer, mrlyweb::walsh_spectrum.

Density theorem boundary

• [Verified] The base-2 D = 4 design census is complete to level 4: 65536 designs close into 402 orbits under the 384 signed coordinate permutations, 400 with k >= 2, 336 both spanning Z^4 and carrying k > 2 so the dimension-above-one density theorem applies to them on its stated sufficient condition, and the 400 eligible canonical representatives realize 189 distinct coprimality sequences (A(1), A(2), A(3), A(4)) with 87 collisions covering 298 classes, the largest being 10 classes on (4, 16, 88, 436) and 10 on (5, 25, 165, 985), by exhaustive enumeration with exact integer minors for spanning and an exact four-coordinate gcd at every point; four terms cannot separate an infinite collision from a short coincidence, and the distribution is over minimum-bitmask representatives rather than unoriented orbits, since coordinate complement moves the arithmetic origin and preserves neither B(F) nor delta nor A(n). Witness: coprime-density-above-dimension-one.
• [Verified] Exact sponge visible census without enumeration, three levels past the feasible: the hybrid A(n) = Sum_(d <= G) mu(d) T*_d(n) - Sum_(gcd > G) S(gcd) with transfer matrix T_d and the big-gcd tail enumerated as multiples in a q^n/g box costs about q^(n(D+1)/2) against enumeration's k^n and gives A(7) = 1038074187, A(8) = 20860210527, A(9) = 418429711224 (22.6 seconds against half a trillion points, the whole ladder in 84 seconds), anchored by the four census terms, direct enumeration at n = 5, 6, and cutoff independence (G = 100 and G = 150 split the work differently and agree on A(9) to the integer). Witness: lab/sponge-visible-census.
• [Verified] The sponge visible census to n = 18 by two engines: admissibility is pairwise disjointness of the digit-1 masks, so A(n) = W(n) - W(n-1) with W(n) = Sum_(m < 3^n, gcd(m,3) = 1) mu(m) (N_n(m) - 1) and N_n(m) a disjoint-triple count over at most 2^n masks of the multiples of m; the engine splits moduli by their multiples count into a closed-form tail, bitset rows, u16 zeta rows and a rank-truncated ranked cube, about 3^n (n 2^n)^(2/3) work, 122.3 s at n = 18 on eight threads with level ratio 4.32, 3.8x its previous form, which it reproduces term for term from A(10) = 8382927031902 to A(18) = 215134797774716879278017, both matching the hybrid census through A(9) and enumeration through A(6); nine counters agree on every modulus to level 8, pinned counters with auto to level 8-11, 38 probed moduli at level 14 and two pinned probes at level 19 cover the u64 cube gate and the u32 rows branch; the new engine alone gives A(19) = 4302768326366633733102921 in 515 s, unwitnessed. Witness: lab/coprime-terms.
• [Proved] The tail of the sponge Mobius sum is closed and admits no hyperbola grouping: for 3^n/2 < m < 3^n, 3 not dividing m, N_n(m) - 1 = 3 + 4 [m has no base-3 digit 1], so the top band of W(n) is three times the Mertens sum over the band's moduli coprime to 3 plus four times a Mertens sum over the base-3 Cantor set; the Y = 3 band is 6 + 7 [mask(m) = 0] + 7 [mask(2m) = 0] + 6 [mask(m), mask(2m) disjoint], every band a Mobius sum over a digit-automatic condition on m, 2m, ..., (Y-1) m; N_n(m) depends on the digits of m, not on floor(3^n/m) (n = 2: m = 5, 8 share the floor with N - 1 = 3, 7; at n = 6 every floor band holding two admissible moduli is non-constant), verified exhaustively at n = 6, 7 against a brute triple loop. Witness: lab/coprime-terms.

Gasket rays and the window

• [Proved] The gasket residual changes coordinate. Every off-diagonal collinear pair of G_n is (sz, tz) for a unique coprime (s,t) and a unique witness z, so R(n) = Sum_z P_n(z) where P_n(z) counts the coprime non-shift pairs one witness realises; the per-pair route needed a constant summable against an active-pair count growing 2.77 a level and is dead by construction, while the per-witness route has its constant. No witness weighs less than 4, hence max(s,t) <= (3^n-1)/8, sharp: the largest multiplier is exactly floor(3^n/8) at n = 4..13. Weight layers scale exactly, R_(3w)(n) = R_w(n-1), because 3 | z_1+z_2 with 3 dividing neither coordinate forces v_3(m z_1) = v_3(m z_2) and the supports collide; checked on all 1869 layers at n = 5..13. Every pair above (3^n-1)/10 carries exactly 4 ordered pairs, its only witnesses being (1,3) and (3,1), verified on all 30028 such pairs at n = 6..13. Witness: gasket-ray-machine, lab/gasket-witness-weights.
• [Proved] The weight-four layer is closed in Fibonacci, the second layer of the residual to close after the shift family. With F_n = {m : (m,3m) in G_n} the no-adjacent-ones set, #F_n = F(n+1) - 1 and R_4(n) = 2 #{(a,b) in F_n^2 : a != b, gcd(a,b) = 1, b/a != 3^j} < 1.0473 phi^(2n), so the whole 3-power orbit obeys Sum_j R_4(n-j) < 1.6945 phi^(2n); exact at n = 4..12 where R_4(n) = 12, 36, 108, 336, 988, 2596, 6672, 17480, 45720, carrying 194096 of R(13) = 863848. Constants safe: 2 phi^2/5 = 1.0472136 and 2 phi^3/5 = 1.6944272. Witness: gasket-ray-machine.
• [Verified] The golden ceiling: M_n(z) <= M_n(1,3) = F(n+1) - 1 for every direction, so the shift ray (1,3) is the heaviest ray of the gasket at every level, and this is the per-witness constant the per-pair route never had. Refutation attempt, briefed to break it: 13158 coprime directions with z_1 <= 120 and z_1 <= z_2 <= 240 at every n <= 40, (1,3) the sole attainer at n = 40; plus six families chosen to favour a breach at every n <= 45 - all binary base-3 pairs below 3^7 (4221 coprime), all no-adjacent-ones pairs below 3^7 (253), all (1,t) with t < 3000 (2998), all consecutive below 1500 (1499), (s,3s-1) (1199) and (s,3s+1) (1199) with s < 1200, 11369 directions in the shelf script and 24088 with the binary family widened to 3^8 in the lab; plus an independent enumeration on a larger box. Zero breaches anywhere. Next rate down is the supergolden 1.4655, the root of x^3 = x^2 + 1, at (1,12), (3,10), (4,9). Witness: gasket-ray-machine, lab/gasket-witness-weights.
• [Proved] Two cheap constructions for the free-digit automaton B(s,t), which used to blow up at large multipliers. It is a constrained tensor square T = S (x) S - U (x) U - V (x) V + W (x) W of a one-coordinate carry automaton with at most (s+1)(t+1) states, so the four-tuple state graph is never built: 729 carry states against 26931 reachable at (365,1094), 81 against 835 at (41,122), agreeing on all 473 coprime pairs below 40 at every level to 9. And at large multipliers the witness box z_1 + z_2 <= floor((3^n-1)/(2 max(s,t))) replaces the automaton entirely in O(W^2 n) digit tests, agreeing on 812 coprime pairs at n = 9 - cheapest exactly where a forward build is most expensive. Witness: gasket-ray-machine.
• [Verified] Conjecture Z evidence to level 17: R(n) = 863848, 2211960, 5549452, 14100688, 35354824 at n = 13..17. R/3^n peaks at 0.8401158 at n = 8 and falls at every level to 0.2737709; R/phi^(2n) peaks at 3.2378233 at n = 12 and falls at five consecutive levels to 2.7724831; the level ratio R(n+1)/R(n) reads 2.5073119 at n = 17, below phi^2 = 2.6180339. Witness: lab/gasket-witness-weights.

• [Verified] Occupied-ray totals differ by exactly 2 between the two counting conventions at every level, 1044842 against 1044840 at n = 13, because one counts the two fibre rays (1,0) and (0,1) and the other does not; nothing else in either table moves, so any occupancy total must say which convention it uses. Witness: lab/dimension-one-ladder; gasket-ray-machine.
• [Verified] The multiplier-count census of the gasket at n = 13: over all 1,044,840 occupied non-fibre rays, 699,508 carry M_n = 1 (17% of Z), 339,530 carry M_n in [2,5] (55% of Z), the ten heaviest rays are exactly the shifts (1, 3^j) and their reverses for j = 1..5 and carry 14% of Z, sum M_n = 1,577,940 = 3^13 - 2^14 + 1 exactly, and max M_13 = 376 = F(14) - 1 reproduces the mass law M_n(3,1) = F(n+1) - 1 from a generator that never mentions Fibonacci, all by exact exhaustive enumeration of the 3^13 gasket points with gcd reduction into a hash table. Witness: lab/dimension-one-ladder; gasket-ray-machine.
• [Proved] The golden partition bound U(z) <= phi^-2 is proved outright on an infinite arithmetic family, not checked direction by direction. Write q = 3^k q_1 for the coordinate divisible by 3 and p for the other. For k = 1 and t = v_3(q_1 - p): U(z) <= phi^-1 (1 - phi^-max(t,2)), so U <= phi^-2 on the whole class k = 1, t <= 2 - 261 of the 360 occupied k = 1 directions of the census - sharply at (1,12) where t = 1 and (3,10) where t = 2, and U < phi^-1 for every such ray but (1,3), the first proof that a whole family of gasket rays grows strictly slower than phi. Refutation attempt: 5422 occupied k = 1 directions picked in the hard corners (deep t, q_1 - p = +-m 3^e for e <= 7, 2q <= p so the far predecessor is live) gave zero violations with equality only at (1,12) and (3,10), the independent first-return series was dominated by the exact solve at 5420 of 5420 checked, and an independent adversary sweep of 910 stratified k = 1 directions in exact Q(sqrt5) found zero violations with equality again only at (1,12) and (3,10); a global U hunt over 17624 occupied directions to weight 6000 found only shift rays above phi^-2. Witness: gasket-ray-machine, lab/gasket-witness-weights.
• [Proved] Occupancy of a gasket ray is a congruence before it is anything else: M_n(z) > 0 for some n forces q_1 = p mod 3, by two lines on last digits with no automaton built - a multiplier m = 3^s m' makes m' p and m' q_1 binary in base 3 and prime to 3, so both end in digit 1. It empties 4588 of the 11691 census directions with 3 | z_1 z_2, and it is only necessary: just 865 of the 7103 matching directions carry mass. Refutation attempt: zero violations over a 400 x 2500 sweep and over both censuses, 1995 occupied directions in the lab universe and 865 on the shelf. Witness: gasket-ray-machine, lab/gasket-witness-weights.
• [Proved] The short first returns of a ray automaton are classified. No first return has length between 2 and v_3(q); f_2 != 0 only at {1,3} and f_3 != 0 only at {1,9}, {1,12}, {3,10}, {4,9}, each equal to 1. Hence U = phi^-2 Sum_(j>=3) f_j phi^(3-j), so U <= phi^-2 says exactly Sum_(j>=3) f_j phi^(3-j) <= 1 and forces f_4 <= 1; the supergolden trio is exactly f_3 = 1 with every later f_j zero. Refutation attempt: a 420 x 2600 sweep over 5281 occupied directions returned exactly those five directions and zero burst failures, and the adversary's exhaustive check to weight 12000 found no fifth f_3 direction. Witness: gasket-ray-machine, lab/gasket-witness-weights.
• [Proved] A two-valued potential read off the out-degrees replaces the exact linear solve: pi = 1 where a live state branches, phi^-1 where it does not, pi(0) = 1, is a super-solution of the golden criterion whenever no branch state has two branching successors, and sweeping it under the same operator gives a decreasing chain of exact Q(sqrt5) upper bounds on U. It settles 849 of the 865 occupied shelf directions - least sweep depth 1 on 760, 3 on 48, 4 on 31, 5 on 7, 6 on 3 - and 1966 of the lab's 1995, reaching 37 and 66 directions outside the branch case. What is left is the 7 shift rays and (1,756), (1,2196), (1,2214), (1,2268), (1,2430), (13,1080), (27,730), (28,729), (40,1053). Refutation attempt: an earlier depth split of 760, 48, 31, 10 was wrong because the sweep skipped depth 5 and the expected tuple had been fitted to that grid, a circular self-check that stayed green; the sweep now runs consecutive depths and the split is the least depth that works. Witness: gasket-ray-machine, lab/gasket-witness-weights.
• [Proved] The golden partition bound restated twice with no automaton in it: Sum_n (M_n(z) + 1) phi^-n <= phi^4 = 3 phi + 2, and equivalently Sum_m phi^-l(m) <= phi over the multipliers m of z, where l(m) is the number of base-3 digits of (z_1 + z_2) m. And the obstruction beyond v_3(q) = 1 is now exact rather than heuristic: the burst forces phi^-2 >= pi(c_0) >= phi^-(k-1) Sum_m pi(q_1 m) over 2^(k-1) burst-floor states of valuation 0 while pi(p) >= phi^-1 at the valuation-0 state p, so any valid potential must separate states of equal valuation by phi^2 (2/phi)^(k-1), which grows without bound - no potential constant on the level sets of v_3, and none constant on the out-degree classes, can work once k >= 2. Refutation attempt: both restatements checked exactly against the linear solve on 111 directions, and the burst identity together with u(p) = phi^-1 checked exactly on all 865 shelf and 1995 lab occupied directions. Witness: gasket-ray-machine, lab/gasket-witness-weights.

Moment ladder and Lemma B

• [Proved] The energy cap E_2K(G_a) <= lambda_2K^a holds at every order with constant exactly 1, which is what turns a ladder rung from a growth rate into a master inequality: the carry box {-r,...,r}^2 with r = floor((K-1)/2) is closed because a digit difference lies in [-K, K] and floor((r+K)/3) <= r for every K >= 1, every walk from the zero state back to itself stays inside S, the strongly connected component of that state, M_S is irreducible by the definition of a component and carries a self-loop at the zero state, hence is primitive with Perron root lambda_2K and positive right eigenvector u, and e_0 <= u/u_0 componentwise with M_S >= 0 gives (M_S^a)_(0,0) <= lambda_2K^a; the attempt to break it looked for the constant C > 1 a reducible matrix would force and found none, since the reduction to the component is free and lambda_2K = lim E_2K(G_a)^(1/a) is the component's own root, checked equal to the full matrix's Perron root at 2K = 4, 6, 8, 10 with the component sizes 1, 7, 7, 19 inside 1, 9, 9, 25 states and the ratios E_2K(G_a)/lambda_2K^a falling monotonically to 1, 0.942327, 0.790590, 0.643725 at a = 6. Witness: lab/dimension-one-ladder.
• [Proved] The master bound at order 2K is one formula for every rung: with a = floor(log_3(p/2)), d_K = ceil(log_3(K/2)), n >= 2a and b = min(a - d_K, n - 2a), Hoelder over three blocks of the digit window of lengths a, a, b at exponents 4K/(2K-1), 4K/(2K-1), 2K gives L_n(p) <= p^2 3^(-((2K-2+kappa)a + kappa_2K b)/(2K)), the outer blocks interpolated between the exact L^2 and L^4 identities and the inner block supplied by the energy cap, both admissible since K 3^(a-d_K) <= 2*3^a <= p; the exponent gain exceeds a exactly when beta < kappa_2K/Lambda_2K with Lambda_2K = 2 - kappa + 2 kappa_2K, which is the rung formula, and the seam 3a against n is the same at every order with the two branches agreeing at n = 3a - d_K, so the feared order-10 crossing does not exist; the adversarial pass tried to break it by hunting a violation over every prime 5 <= p <= 199 and 2 <= n <= 24, running the order-10 block alone in its 833 applicable cases as well as the min over all orders, and found none, worst ratio 0.7839 at (p, n) = (11, 2) for the order-10 block alone and 0.8755 at (13, 4) for the min, and by hunting an uncovered or negative-gain case in the main range 3a > n over eta in 0.001..0.1 and n = 6..400, finding none, worst geometric sum over cap 0.8630. Witness: lab/dimension-one-ladder.
• [Proved] The order-10 rung is unconditional and the exponent needs no root-finding to be trusted: with Lambda_10 = 4.436585106, main-range decay Lambda_10/10 = 0.443658511 and geometric constant 2/(1 - 3^(-Lambda_10/10)) = 5.1842 rounded to 6, the ladder reads Sum_(z < p <= 3^((beta_0^(10) - eta) n)) T*_p(n)/3^n <= 2/z + 35 z^(1-kappa) + 40 * 3^(-(kappa-1)n/8) + 6 * 3^(-0.443658511 eta n) for eta in (0, beta_0^(10)), z >= 5, n >= 1, the first three terms being the unchanged order-4 bookkeeping; a Sturm count on the exact quartic x^4 - 7833x^3 + 7916949x^2 - 850684437x + 13054946580 places no root above 66641136626/10^7 and exactly one root in the bracket of width 10^-7 below it, so lambda_10 < 6664.1136626, kappa_10 > 1.985805792698 and beta_0^(10) > 0.447597813453, every digit truncated down, never rounded, so the short form printed everywhere is 0.4475978; rows 12 through 20 stay Conjecture for a different reason, their lambda_2K being floating eigenvalues and not certified algebraic numbers, so the energy cap alone does not promote them. Witness: lab/dimension-one-ladder.
• [Proved] The dimension-one moment ladder has a tenth rung and the lower wall is 0.4475978, not 0.446717: the exact 25-by-25 integer carry matrix M_10 on the box {-2,-1,0,1,2}^2 satisfies E_10(G_a) = (M_10^a)_((0,0),(0,0)) with first energies 1, 4653, 28967859, 190911254427, 1270015973323281, 8461182216374750493 (matched by direct convolution of the digit set at a = 1, 2, 3), its characteristic polynomial factors symbolically as x^6 (x-120)(x^2-450x+12231)(x^3-2190x^2+282096x-5186835)^2 (x^3-990x^2+116154x-2569725)^2 (x^4-7833x^3+7916949x^2-850684437x+13054946580), the Perron root is the largest root of the quartic lambda_10 = 6664.113662506, so kappa_10 = 1.985805792712 and beta_0^(10) = kappa_10/(2 kappa_10 + 2 - (3 - log_3 5)) = 0.447597813454, above the eighth rung by 0.000880502992, with Holder block exponents 20/9, 20/9, 10; "dimension one" means the similarity condition log_q |F| = 1 at base 3 and not the base-2 gasket of density 16/(3 Pi^2), and the order-10 three-block master bound with explicit constants and checked regime seams is now written, so the rung is a theorem and a re-proof of target-uniform Lemma B at that edge. Witness: lab/dimension-one-ladder; lemma-b-pincer.

Kronecker word order

• [Verified] Boundary is order-blind at word length 2 and order-sensitive from length 3: on the 4 x 4 grid only the four central cells can be interior and their requirements pair up under the factor swap, (S_1, S_4) against (S_4, S_1) and (S_2, S_3) against (S_3, S_2), so interior and hence boundary is symmetric in the two factors, exact on all 256 code pairs with the interior formula matching direct computation on all 256; at length 3 it fails on 36 of 210 multisets over the ten-code library of every code of fill 2 or 3. Witness: lab/magic-words.
• [Verified] Connected components of a mixed Kronecker word are order-sensitive, with minimal witness the multiset {3, 6}: comp(A_3 (x) A_6) = 4 against comp(A_6 (x) A_3) = 2, both factors of fill 2, because the inner tile's contacts decide whether adjacent outer copies merge (A_6 is two isolated cells with no boundary contact, so four cells stay apart, while A_3 is a connected vertical pair sitting in two non-adjacent outer copies), enumerated by two independently written renderers, the study's substitution pass and the crate's Kronecker factory, agreeing cell for cell. Witness: lab/magic-words.
• [Proved] The k = 2 designs at base 2, D = 2 split into adjacent (codes 3, 5, 10, 12) and diagonal (codes 6, 9), and commutativity of the component count follows the split: adjacent times adjacent commutes, diagonal times diagonal gives 4 in both orders, and adjacent times diagonal never commutes, always 4 against 2, by contact geometry, the diagonal pair having neither a face-adjacent cell nor a contact in either direction, and on all 15 pairs among the six codes with zero violations. Witness: connectivity.md, lab/magic-words.
• [Verified] Any two designs with k >= 3 at base 2, D = 2 commute and give exactly one component in either order, since every such tile is connected and carries both a top-bottom and a left-right contact, so every pair of adjacent outer copies merges through the inner tile and connectedness of the outer tile merges them all; exhaustive over the four k = 3 tiles plus the full tile, all 10 pairs among codes 7, 11, 13, 14, 15. Witness: lab/magic-words.
• [Proved] Block reduction: every periodic schedule (c_1, ..., c_p)^L equals the L-fold self-similar product of its one-period composite tile A_(c_1) (x) ... (x) A_(c_p), of base prod_i n_i and fill prod_i k_i, by associativity of the Kronecker product alone, so periodic mixed words carry no new theory and the first genuinely non-stationary behaviour requires an aperiodic word; cell for cell on six test cases at periods 2 and 3 and lengths to 6, the study's flat rendering against Tensor::fractal of the composite. Witness: dimensions.md, lab/magic-words.
• [Verified] Contact counts of a mixed Kronecker word are exactly multiplicative, h(A_w) = prod_i h(A_(c_i)) and v(A_w) = prod_i v(A_(c_i)) for the row and column contact counts, because L(A (x) B) = L(A) (x) L(B) on outer columns and rows and the inner product of Kronecker products is the product of inner products, so whether adjacent copies touch is order-blind and decided factor by factor even where the component count is not; induction on length, exact on all 15^3 words with zero mismatches, the correct strengthening of the length-2 boundary theorem. Witness: lab/magic-words.
• [Verified] The naive geometric transfer state for mixed products is unbounded: kappa(A_w), the number of components meeting a contact position and so still able to merge with a neighbouring copy, reaches 2^(L-1) on the family w_L = (15^(L-1), 3), whose product is 2^(L-1) disjoint full-width rows each meeting both the left and right column, from the full-tile Kronecker power, exact at L = 2..10, with exhaustive search over all 15 codes showing w_L is a maximiser at L <= 4 with maxima 1, 2, 4, 8; the same family's component count is exactly 2^(L-1) too. Witness: lab/magic-words.
• [Proved] The component count of a mixed Kronecker word is a rational series of Hankel rank 4: with lambda = (1,0,0,0), gamma = (1,1,1,1)^T and one 4 x 4 integer matrix per code in six classes, comp(A_w) = lambda M_(c_1) ... M_(c_L) gamma at every word, by the four transfer laws in the observable frame (gamma, h, v, phi) and induction from (comp, H, V, fill)(A_e) = (1,1,1,1); the same induction reads H, V and the fill off the same matrices. Witness: connectivity.md THE JOINT SPECTRAL RADIUS OF THE COCYCLE, lab/jsr-schedules.
• [Verified] Euler characteristic, boundary and holes of a mixed Kronecker word are rational series in the word, of Hankel rank 4, 8 and 11: there are lambda, gamma and one matrix per code with phi(A_w) = lambda M_(c_1) ... M_(c_L) gamma, exhaustive on all 54240 words of length at most 4 over the 15 non-empty codes plus 120 seeded words of length 5 to 7, all four observables, zero mismatches, built by Hankel-basis elimination in exact rational arithmetic; the rank is unexplained, staying 4 while the geometric state grows like 2^(L-1) and the component count reaches 2 * 4^(L-1) on (15^(L-1), 6), the checkerboard and the largest component count any subset of the 2^L grid can carry. Witness: lab/magic-words.
• [Verified] Order sensitivity of mixed products is exactly the noncommutation of the cocycle matrices: 14 of the 15 pairs of component-matrix classes fail to commute, the only commuting pair being the two zero-contact classes, k = 1 and the diagonal pairs, whose matrices have rank 1 with M_(6,9) = 2 M_(1,2,4,8); the vertical and horizontal domino classes carry different matrices, so the representation sees more than the square's full symmetry group; checked on all 15 class pairs. Witness: lab/magic-words.
• [Proved] The scale dimension of a magic word over a finite alphabet is a frequency functional: log side and log fill of a prefix are sums of per-letter values from a finite set, so whenever each letter's frequency exists the ratio d_L converges to the frequency-weighted average, and in particular every uniquely ergodic word (Thue-Morse, period-doubling, Fibonacci) has a scale dimension equal to that average; open are finite alphabets without letter frequencies and all unbounded alphabets, where frequencies can exist while the dimension oscillates. Witness: magic.md.
• [Proved] The fill assumption lab/slice-ladder-controls states before printing its five staircase dimensions is discharged: the parity-carpet code at odd side q fills E^2 + 2EO = q^2 - ((q-1)/2)^2 with E = (q+1)/2 and O = (q-1)/2, the octagonal fill 3k^2 - 2k at q = 2k - 1 already on this ledger for the same rule, so the staircase dimensions stand without the assumption. Witness: magic.md, sequences.md.
• [Proved] The hyperoctahedral group acts diagonally through the Kronecker product, g . (A (x) B) = (g . A) (x) (g . B), because reflecting a mixed-radix coordinate reflects every digit at once, so a magic word canonicalises under one shared symmetry applied to all letters and never letter by letter. Witness: magic.md.
• [Proved] Every value in [0, log 8 / log 3] is the scale dimension of some word over the two letters carpet(3) and c8(3): rational carpet frequencies by periodic words, irrational by Sturmian words, endpoints by constant words, all through the frequency functional. Witness: magic.md.

The component exponent

• [Proved] The component count of a two-letter magic word has an exact closed form on 59 of the 105 letter pairs at base 2, D = 2, namely 9 of the 15 pairs of distinct symmetry classes and all 6 pairs inside one class: 2^(k-m) with k the last unit place and m the unit count for a unit against a domino (16 pairs), 2^(number of diagonal letters) for a unit against a diagonal (8), 2^(L-r) with r the terminal run for two dominoes of unlike orientation (4), 2^k with k the last diagonal place for a domino against a diagonal (8), 2^(n-j) with n the number of full letters and j their terminal run for a domino against the full tile (4), 1 for the gasket class against the full tile (4), and 1 inside a class except 2^L inside the diagonal class (15); proved by the zero-contact cut on the first four families, again by a rank-1 telescope on the domino against the diagonal, by the row-block argument on the domino against the full tile, and by the contact split with the k >= 3 line on the rest; exhaustive per pair on all 32766 words of length at most 14 against the representation and all 254 words of length at most 7 against the drawn cells, zero mismatches. The other 46 pairs are closed by the suffix recursion below. Witness: lab/magic-words, connectivity.md.
• [Proved] The zero-contact cut: since contacts multiply, a suffix has h = v = 0 as soon as it holds one letter with h = 0 and one with v = 0, adjacent copies of that suffix tile can never merge, and comp(A_w) = fill(A_prefix) * comp(A_suffix) at the last such suffix; 49420 of all 54240 words of length at most 4 admit the cut with zero mismatches. Its scope is part of the law: the other 4820 words carry nonzero contact in one direction at every suffix, and every word over a domino and the full tile is among them, so that family needs the row-block argument instead. Witness: lab/magic-words, connectivity.md.
• [Proved] The row-block law, and the one non-trivial exponent found: over a domino and the full tile the filled set is a product of a row set with the whole column range, |R| = 2^n at n full letters, each row of R is a full line, and two rows of R are adjacent exactly inside a block of 2^j rows at j the terminal run of full letters, so comp = 2^(n-j); at equal letter frequencies the exponent is (log 2)/2 against per-letter values 0, 0 and a fill ceiling of (3/2) log 2, strictly between them, which no other named pair achieves. Witness: lab/magic-words, connectivity.md.
• [Proved] On every one of those 59 pairs, along any word in which both letters occur with positive frequency, the component growth rate exists, is a function of the letter frequency vector alone, and is read off the closed form; existence is earned from the formula and not assumed. The hypothesis is load-bearing and the same statement without it is false: over {3, 6} at frequency vector (1, 0) the constant word 3^L has rate 0, the word carrying the diagonal letter at the square places has rate log 2, and the word carrying it at the powers of 2 has upper rate log 2, lower rate (log 2)/2 and no limit, the last two having orbit closures that are uniquely ergodic but not minimal. Witness: lab/magic-words, connectivity.md.
• [Proved] The constant word is a degenerate probe for the component exponent: comp(A_(c^L)) = comp(A_c)^L with comp(A_c) = 2 exactly on the diagonal class, so the one linear functional exact on constant words is Phi(f) = (f_6 + f_9) log 2, and it fails at every interior frequency on 7 of the 9 named class pairs, holding only for a unit against a diagonal and for the gasket class against the full tile; the failure is a wrong shape rather than a wrong coefficient, since the true exponent extends to no vertex. The caveat travels with the result: on five of the seven, 28 of the 32 letter pairs, the exponent equals the fill exponent and so saturates the trivial ceiling comp <= fill, carrying nothing the order-blind fill law did not already give. Witness: lab/magic-words, connectivity.md.
• [Verified] The Thue-Morse run structure that makes the exponent exact: no three equal letters in a row, so every terminal run has length at most 2 and every prefix rate is within 2/L of its limit; exactly L/2 of each letter at every even length, not merely in the limit; over all lengths to 2^20 the terminal run of one letter takes the value 0 on 524288 prefixes, 1 on 349526 and 2 on 174762; the first 2^20 letters hold 349525 complete runs of length 1 and 349525 of length 2 with one unfinished run at the cut; and the run-boundary word t_n xor t_(n+1) is the period-doubling word on all 1048575 terms. Witness: lab/magic-words.
• [Proved] The remaining 46 letter pairs carry exact closed forms too, so all 105 are solved, and the mechanism is a suffix recursion rather than anything spectral: a heavy suffix letter, meaning one of the five connected codes 7, 11, 13, 14, 15 that carry both contacts, leaves the count unchanged because the block graph of A_w (x) A_c is isomorphic to the cell graph; a zero-contact suffix letter, meaning one of 1, 2, 4, 8, 6, 9, leaves isolated cells and collapses the count to a fill; and a domino suffix letter turns the count into the number of maximal runs, whose recursion H(A_wq) = H(A_w) + fill(A_w) at a gasket letter telescopes. Hence comp(A_w) = fill(A_(w_1..p)) at p the last zero-contact place on the 30 pairs of a heavy letter against a light one, giving 3^(g-j), 2^d 3^(g-j), 4^(F-j) and 2^d 4^(F-j) with j the terminal heavy run, and comp(A_w) = 1 + sum of fill(A_(w_1..i-1)) over the gasket places i at or before the last domino place m on the 16 gasket-against-domino pairs, (3,7) and (5,7) among them, the eight column-domino pairs being the transpose of the eight row-domino ones; exhaustive on all 753572 words of length at most 13 against the representation and all 11684 of length at most 7 against the drawn cells, zero mismatches. Witness: lab/magic-words, connectivity.md.
• [Proved] At every interior letter frequency the component exponent exists, is order-blind and equals the fill exponent f_a log k_a + f_b log k_b on all 46 of those pairs; existence and order-blindness hence hold on all 105, saturation on 89 of them; the interior hypothesis is used exactly once, to force the terminal run of the heavy letter to be o(L), since a run of eps L would freeze the other letter's count on that block and contradict its positive frequency, and the gasket-against-domino case runs through the sandwich T < comp(A_w) <= 1 + (3/2) T at T = fill(A_(w_1..i*-1)) and i* the last gasket place at or before the last domino place, measured in [1.0004, 2.0000] on six named words with no violation. Nothing is claimed for a word whose letter frequencies fail to exist. Witness: lab/magic-words, connectivity.md.
• [Proved] The two rates the study could not identify are identified exactly: along the Thue-Morse word over any of the 16 gasket-against-domino pairs, (3,7) and (5,7) included, the component exponent is (1/2) log 6 under either letter reading, with the two-sided certificate |log comp(A_(w_1..L)) - (L/2) log 6| <= log 108 + (1/2) log(3/2) < 4.885 at every L >= 4, because the word is cube-free, which caps the sandwich suffix at fill/T in [6, 108], and balanced, which pins |n_q(L) - L/2| <= 1/2; the study prints the value as 0.895879734614027 nats, a float labelled as such. Measured to L = 2^14 on all 16 pairs and both readings the largest deviation is 4.273459 nats against the certificate's 4.884864. Witness: lab/magic-words, connectivity.md.
• [Proved] The frequency functional Phi(f) = (f_6 + f_9) log 2 is refuted at every interior frequency on 78 of the 105 letter pairs and exact on 27, the 46 new pairs being refutations to a pair; and the exponent saturates the trivial ceiling comp <= fill on 89 of the 105 and falls short on 16, so the domino against the full tile is the unique class pair on the whole alphabet, and not merely among the named 59, whose exponent sits strictly between the constant-word values and the fill ceiling. The comparison is against the value of Phi and never against a periodic word. Witness: lab/magic-words, connectivity.md.
• [Proved] The interior hypothesis is sharp on a pair carrying no diagonal letter, so the pathology is not a property of the diagonal class: over (3, 7) at frequency (1, 0) the constant word 3^L has rate 0, the gasket at the square places gives log 2, and the gasket at the powers of 2 has upper rate log 2, lower rate (log 2)/2 and no limit, with comp pinned to fill(A_(w_1..i*-1)) at i* = 2^k throughout (2^k, 2^(k+1)], so the accumulation set of the prefix rate is the whole interval [(1/2) log 2, log 2] and not its two endpoints. Witness: lab/magic-words, connectivity.md.
• [Proved] A common invariant cone exists and the gasket-against-domino pair is primitive in it, and none of it is needed: phi = (1,2,2,4)^T is a common right eigenvector with M_c phi = k_c phi, so it normalises the row orbit and only that orbit, and in the resulting chart comp/fill = 1 - b - c with the letters acting by N_gasket(b,c) = ((1+b)/3, (1+c)/3) and N_domino(b,c) = ((1+b)/2, 0); the set {0 <= b <= 1, 0 <= c <= 1/2, b + c <= 1} is invariant under both and the length-3 word gasket-domino-gasket maps it strictly inside, vertex images (5/9,1/3), (11/18,1/3), (5/9,1/3), (7/12,1/3) with b + c at most 17/18, three being minimal since domino-gasket sends (1,0) to (2/3,1/3) on the face. Witness: lab/magic-words, connectivity.md.
• [Verified] A by-product and a smoothness split: comp(A_((7,3)^k)) = (6^k + 4)/5, reading 2, 8, 44, 260, 1556, 9332 and checked to k = 8 against both the closed form and the representation, a stationary control whose per-letter rate is (1/2) log 6 again; and since every closed form on the other 89 pairs gives a count of the form 2^a 3^b, the gasket against a domino is the only family whose counts are not smooth, the largest at L = 8 over (3, 7) being 1094 = 2 x 547. Witness: lab/magic-words, connectivity.md.

The tile monoid

• [Proved] Factorisation of a 0/1 tile is unique once the ordered side profile is named: if A (x) B = A' (x) B' with A, A' of side m and B, B' of side n, all non-empty, then cutting the composite into an m x m array of n x n blocks reads A off as the 0/1 indicator of the non-zero blocks and B as any one of them, since every non-zero block equals B and B is not the zero tile, so A = A' and B = B'; the non-empty tiles under the Kronecker product are therefore a monoid graded by side, cancellative and atomic, whose block test decides factorability at a named shape in O(N^2) of exact integer comparison, the 0/1 hypothesis being load-bearing since over the rationals A (x) B = (kA) (x) (k^-1 B), and the block reading being the 0/1 shadow of the Van Loan and Pitsianis rearrangement. This is a rediscovery and is cited, not claimed: it is Lemma 2.4 and section 2 of Voet and De Novellis, Identifying Kronecker product factorizations, arXiv:2510.25292, for binary matrices under equality. Witness: lab/code-factorisation, magic.md, arXiv:2510.25292.
• [Proved] The ordered side profile is not recoverable from the composite, so the tile monoid has no unique factorisation, and the failure is not axis-separable: [6]{(0,0),(2,2)} is both c1 (x) c257.q3 and c17.q3 (x) c1, all four letters of prime side and hence irreducible with differing multisets, and the tile is not a rectangle; the mechanism is an infinite family rather than a side-6 accident, since I_m (x) I_n = I_mn = I_n (x) I_m and E_m (x) E_n = E_mn = E_n (x) E_m at every pair of sides by the symmetry of (nm - 1) - x = (n - 1 - i) m + (m - 1 - j) in m and n, the same digit identity the diagonal action already carries, so taking m and n distinct primes gives four irreducible letters at every side with two distinct prime factors; the existence of shape-distinct factorisations is Example 2.5 of Voet and De Novellis, and new here are the axis-separable refutation and the I/E family. Witness: lab/code-factorisation, magic.md, arXiv:2510.25292.
• [Proved] Neither the length nor the side multiset of a factorisation is an invariant of the composite, first at side 12 and at no smaller side: [12]{(0,0),(3,3)} reads as three irreducible letters of sides 2, 2, 3 and as two of sides 3, 4, the side-4 letter [4]{(0,0),(3,3)} being irreducible because its one candidate cut has two unequal blocks, while every side below 12 is a prime power, where factorisation is unique, or a product of two distinct primes, where every letter has prime side; the short reading needs a letter of composite side, which no plane code is, so length and the side multiset are invariants for free inside the magic-word submonoid generated by prime-side letters and fail in the full monoid, and the alphabet of the words is not the alphabet of the monoid, 65310 of the 65535 non-empty side-4 tiles being irreducible already and the reducible share falling from 0.343328% at side 4 to 0.0000221% at side 6; a factorisation whose sizes are not all prime is Example 2.6 of Voet and De Novellis, and new here are the minimality of side 12 and the alphabet gap. Witness: lab/code-factorisation, magic.md, arXiv:2510.25292.
• [Proved] Two factorisations of one tile admit a common refinement exactly when the union of their cut chains is totally ordered by divisibility, because a cut at d' dividing a cut at d factors the side-d left factor through the side-d' one; so unique factorisation holds at every prime-power side, where the divisors are a chain, and fails exactly when the cut set L(C) holds two incomparable divisors, checked with zero mismatches against direct enumeration of every irreducible factorisation over all 339795 side-12 plane-code composites, of which 7023 carry two or more factorisations and 2376 carry factorisations of unequal length. Witness: lab/code-factorisation, magic.md.
• [Proved] The tile monoid is not a trace monoid, so no canonicalisation may sort or commute letters: [2]{(0,0)} (x) [3]{(1,1)} = [6]{(1,1)} = [3]{(0,0)} (x) [2]{(1,1)} uses four pairwise distinct irreducible letters, which no commutation of a letter pair can produce, and only 11 of the 171 side-6 cross-shape tiles are honest commutations against 160 rewritings. Witness: lab/code-factorisation, magic.md.
• [Proved] Two letters render one tile at one side only at side 3: if a base-2 code and a base-3 code agree cell for cell at side n then row r equals row r' whenever r = r' mod 2 or mod 3, and at n >= 4 those two partitions join the whole row range, so every row and every column agrees and a non-empty constant tile is the full tile; the census is 480 pairs at side 2, 15 at side 3 and the full tile alone at sides 4, 5, 6, 7, 8, 9, 12 and 18, the carpet's side-3 partner is uniquely c495 of fill 8, and at side 9 the readings separate into fills 65, 72 and 64 on three pairwise distinct tiles, though read as level-2 fractals of the side-3 letter they do not diverge at all, since at side 3 they are one tile. Witness: lab/code-factorisation, magic.md.
• [Proved] The canonical name of a composite is the code together with its ordered side profile, c<code>(n_1 x n_2 x ... x n_L), and the two non-injectivities are different objects that must be disambiguated in order: the render collision belongs to the alphabet alone and tabulates once per base and side, since fixed-shape uniqueness proves the fold never creates one, while the fold collision belongs to the profile; profiles of different length occur, so the tie-break orders profiles by length first, finest before coarsest, then lexicographically, before the diagonal-action class rep breaks what is left. Witness: lab/code-factorisation, magic.md.
• [Verified] The side-6 census: the two shape images are injective at 7665 tiles each, 171 tiles lie in both, so 15159 of the 2^36 - 1 side-6 tiles are reducible once the overlap is removed and 68719461576 are irreducible; of the 171, 121 are axis-separable and 50 are not, 11 are commutations and 160 rewritings, fills run 1:36 2:64 3:32 4:16 6:14 12:8 36:1 over the 171 and 2:16 3:32 6:2 over the 50 with outer-fill signature (1,1):24 (1,2):8 (1,3):16 (2,3):2, and the 48 that are neither separable nor commuting are exactly the 48 carrying a one-cell letter in at least one reading and exactly the 48 carrying a one-cell outer factor in at least one reading, three statistics on one set checked as sets rather than as counts, since the two one-cell readings differ elsewhere (0:23 1:8 2:140 against 0:23 1:60 2:88 over the 171). Witness: lab/code-factorisation, magic.md.
• [Verified] 121 = 11 x 11 is arithmetic with a checked bijection: the 121 axis-separable side-6 cross-shape tiles are exactly the products R x C of the 11 lines that factor in both radix orders, {0} {1} {0,1} {2} {0,2} {3} {4} {5} {3,5} {4,5} {0..5}, verified as set equality and not as a count. Witness: lab/code-factorisation, magic.md.
• [Verified] Counting reducible tiles at prime-power side is inclusion-exclusion over the divisor chain, equivalently the series I = T/(1+T) on the grading, giving 225 at side 4, 1962675 at side 8, 261121 at side 9, 553402322215537199175 at side 16 and (2^25 - 1)^2 = 1125899839733761 at side 25, cross-validated in one dimension against exhaustive brute force at N = 4, 8, 16, 9 reading 9, 63, 1431, 49; nothing new happens at a prime-power side, where the two side-8 shape images of 983025 tiles each meet in exactly the 3375 triple products of base-2 codes, checked as set equality, so 3375 is pure associativity and never stands beside 171. Witness: lab/code-factorisation, magic.md.
• [Proved] One-cell letters commute exactly when a(n - 1) = b(m - 1), giving gcd(m - 1, n - 1) + 1 singleton pairs per axis and, where no common power exists, gcd(m - 1, n - 1) + 2 commuting pairs in one dimension, checked at nine side pairs and exceeded only at (3,9) at 7 against 4 through the common-power branch; at base 2 against base 3 this gives the 11 commuting code pairs (1,1) (2,4) (3,7) (4,64) (5,73) (6,84) (8,256) (9,273) (10,292) (12,448) (15,511), nine of them a commuting row line against a commuting column line and the other two the diagonal and the antidiagonal, with base-2 codes 7, 11, 13, 14 unpartnered, so the carpet code itself does not commute. Witness: lab/code-factorisation, magic.md.

Half-ball chords

• [Verified] The half-disk chord constant (Zerr) decomposes into an integer and an area: P(the chord through two uniform points of the upper unit half-disk crosses the diameter) = I_diam/(3 Area(H)^2) by Blaschke-Petkantschin, the flat-face chord-cube integral is the integer I_diam = 4 and Area(H)^2 = Pi^2/4, giving 16/(3 Pi^2); the whole computation collapses to Integral_(-1)^(1) (-a u + sqrt(1 - a^2 + a^2 u^2))^3 du = 2 for every a, the even part of the cube being the exact derivative d/du [u (1 - a^2 + a^2 u^2)^(3/2)] with R(+-1) = 1, checked by the exact derivative, by differentiation under the integral in a, at 50 digits on 50 values of a, and with a symbolic residual of exactly zero at every step. Witness: lab/half-ball-mismatch.

Conjecture S: odd half

• [Proved] The slice census has the trigonometric product formula P(e^(i psi)) = e^(i D psi) (2 cos psi)^(D-1)(D + 2 cos psi), and the sheaf census b(L) (coordinate sum == D(3^L-1)/2 mod 3^L, equally the free-end carry count) is b(L) = 3^(-L) sum_(m<3^L) prod_(j<L) Phi(2 pi m 3^j/3^L) because the extraction phase is the accumulated palindromic phase; every unit tower ends at 2 pi u/3 with factor (-1)^(D-1)(D-1), so at odd D the integrand is pointwise nonnegative, b(L) >= (fill/3)^L and rho_D >= fill/3, and det(fill I - 3 M_even) == fill^n mod 3 gives rho_D > fill/3 strictly at every odd D = 0, 2 mod 3 and through D = 80 in the class 1 mod 3 by exact determinants (Bareiss, three 61-bit primes and Berkowitz agreeing); the bijection is brute-forced at D = 2..8, L <= 4 and the phase cancellation matched to 50 digits at D = 2..12. Witness: slice-recurrence-order.
• [Proved] The pinning |rho_D - fill/3| <= 2(D-1)/3 holds unconditionally (even D in [fill/3 - 2(D-1)/3, fill/3 + (D-1)/3], odd D mirrored) because the core's column sums take exactly the values fill/3 + 2 eps and fill/3 - eps; exactly 3 rho_D = fill + (-1)^(D-1)(D-1)(3 p_D - 1) with p_D the Perron carry vector's mass on carries divisible by 3, well defined since the core is irreducible for all D; so dim_slice - (d-1) -> 0 like D^2 2^(-D) regardless of sign, and Conjecture S entire is the parity-free inequality p_D > 1/3; checked by power iteration at D = 2..20 and entrywise column sums at D = 2..80. Witness: slice-recurrence-order.
• [Verified] The sign-law mechanism is universal: at every base q >= 3 and u != 0 mod q the design symbol has g_q(2 pi u/q) = -1, the full digit sum vanishing at a nontrivial q-th root of unity and the middle digit contributing 1, so the innermost tower factor is (-1)^(D-1)(D-1) at every odd base and the mechanism is q-th-root evaluation, never P(-1); the odd-D inequality rho >= fill/q travels with scope D >= -min g_q (9/4 at q = 5, (34+14 sqrt 7)/27 at q = 7, growing like 0.217 q), strict when D != 1 mod p for some prime p | q; the sign law is exact by Sturm counts at base 5 D = 2..26, base 7 D = 2..18, bases 9, 11 D = 2..12 and (q,D) = (21,3), (31,5), (51,5), (101,3). Witness: slice-sign-even-half.

Conjecture S: the even-half transient

• [Verified] The two-step census contraction 9 b(2j+2) <= fill^2 b(2j) with 3 b(2j+1) < fill b(2j) has zero violations through index 400 at every even D <= 50, margin peaking near (D-2)/(D+2), and in exact integers at even D = 2..30 to index 40 the margin sits strictly below (D-2)/(D+2) and rises toward it, 0.8704914 against 0.875 at D = 30; with irreducibility and nonvanishing it would close the even half. Witness: slice-recurrence-order.

The parity-carpet stack spectrum

• [Proved] The flat odd-scale parity-carpet stack's spectrum is the divisor field of the frequency gcd and nothing more: the sine coefficient of G_L at odd (a,b) is (1/(pi^2 ab))[1 - sigma_1^S(a)/L - sigma_1^S(b)/L + sigma_2^S(gcd(a,b))/L] and vanishes at any even index, the interaction part carrying exactly sigma_2(gcd(a,b))/(ab); Parseval splits the variance blockwise into the two terms of the carpet law, re-proving the moire variance formula; every spectral statistic is an Estermann-Ramanujan zeta quotient, sum sigma_2(gcd)(ab)^(-w) = lambda(w)^2 lambda(2w-2) and sum sigma_2(gcd)^2 (ab)^(-w) = lambda(w)^2 lambda(2w-2)^2 lambda(2w-4)/lambda(4w-4) with lambda the odd zeta; a stack weighted n^(-s) renders sigma_(2-s) as its spectrum; divisor information only, no new L-function; coefficients checked cell-exactly at L = 14 to 47 digits and Parseval against the exact rational variance; the object is the flat odd-scale stack, not the all-scales Farey stack. Witness: moire-correlation-laws.

Conjecture S: even half

• [Proved] The reduction chain for Conjecture S runs at every odd base q, middle-digit design, D >= 2: the digit polynomial is palindromic with strictly positive support; the transfer recursion hat u_(L+1)(psi) = (1/q) sum_r Phi(y_r) hat u_L(y_r) holds with exact phase cancellation (the palindromic centre is the carry offset Dm); the step identity is W_k = (-1)^(D-1)(D-1) q^(k-1) V(k-1) with V(L) = q m0(L) - b(L); the carry core has reachable set exactly {|c| <= floor((D-1)/2)}, is irreducible and aperiodic, and its Perron root is rho_D; eventual contraction (V(L) >= 0 for all L >= L_0, even D) implies rho_D <= fill/q; and det(fill I - q M_even) == fill^n mod p for any p | q, primality unused, gives strictness whenever p nmid fill; the Perron-Frobenius asymptotic is unnecessary (b(L) >= (M^L)[0,0] suffices) and the base-3 contraction hypothesis weakens to its eventual form since the odd-L dip is a finite transient. Witness: slice-sign-even-half.
• [Proved] The exact two-step reduction 9 b(2j+2) - fill^2 b(2j) = -(D-1)[fill V(2j) + 3 V(2j+1)], the two-step weight identity C_2(y) = g_5(2y) (the base-3 two-step symbol is the base-5 symbol, the comb constant having minimal polynomial x^3 - 9x - 9), and the orbit identities G(3t) = cos(t) G(t) and Ntilde(3^a pi) = ((D-2)/(D+2))^a Ntilde(pi). Witness: slice-sign-even-half.
• [Proved] The Collatz-Wielandt certificate closes the even half per dimension at any odd base, strictness included: B >= 0, x > 0, Bx < theta x componentwise imply rho(B) < theta with no irreducibility needed; with beta_K = (M^T)^K 1 > 0 (positive by column-sum positivity alone, colsum(c) = (fill + (-1)^(D-1)(D-1)(q[q|c]-1))/q > 0), if q beta_(K+1)(c) < fill beta_K(c) for every |c| <= (D-1)//2 then rho_D < fill/q strictly, bypassing the mod-p determinant lemma and every exceptional class; the base-5 mass identity on the right Perron vector of the full core is 5 rho_D = fill - (D-1)(5 p_D - 1), so the even half is p_D > 1/5, and the left-vector reading is false at D = 8 (p_LEFT = 0.1428 < 1/5); K = 2 certificates are exact at D = 16, 30, 44, 60. Witness: slice-sign-even-half.
• [Verified] The row certificate v^T M^t >= 0 is sound - V(L) = sum_j (v^T M^t)_j u_(L-t)(j) with both factors nonnegative, so one integer t with v^T M^t >= 0 entrywise plus the exact prefix V(0..t-1) >= 0 proves V(L) >= 0 for all L, monotone in t - and at base 5 its minimal depth is t(D) = max(1, ceil(log_5(2D-3)) - 1), breakpoints exactly at R = (5^(k+1)-1)/4, checked to even D = 400 with fresh rows at 150, 250 and the boundary 314|316; the 782 at k = 4 is an extrapolation unobservable below D = 1566. Witness: slice-sign-even-half.
• [Proved] The 2-adic strictness lemma, complementary to the mod-p lemma: det(fill I - q M_even) == (-q)^n det(M_even) (mod fill) by principal-minor expansion (every k < n term killed by fill^(n-k)), so a prime p | fill, p nmid q with v_p(det M_even) < v_p(fill) forces det(fill I - q M_even) != 0, that is rho_D != fill/q, which upgrades a certificate's <= to <; the two lemmas' silent classes (p | q against p | fill) are complementary; at base 5 v_2(fill) = 2(D-1) + v_2(D+4) while v_2(det M_even) <= 26 out to D = 156, so the test holds everywhere including the exceptional class D == 6 (mod 10) to D = 156, and at base 3 the class D == 4 (mod 6) to D = 118; the 5-adic side is large and erratic and the mod-25 angle is dead; open: a uniform bound on v_2(det M_even), <= n sufficing for all even D >= 4. Witness: slice-sign-even-half.
• [Proved] The even half of Conjecture S at base 5 is a theorem entire: on the base-5 middle-digit solid rho_D < fill/5 for every even D >= 2, hence dim_slice < d - 1 at every even D - the Fourier form beta_K(c) = 5^(-K) sum_n F_K(n) e^(2 pi i n c/5^K) (the product-formula phase cancellation iterated), the exact telescoping fill beta_K - 5 beta_(K+1) = 5^(-K)(D-1) Sigma_K(c) with the fill-power leading terms cancelling identically, the frequency-separation lemma Q_K(n)/Q_K(1) <= 0.768 for every n != +-1 (three-branch residue analysis in rigorous intervals, maximum 0.7679580, read 0.7679541 in an earlier check, the j >= 3 branch exhaustive at j = 3, 4, 5), and the criterion at depth K(D) = Theta(log D), analytic for even D >= 18 (K(D) >= 2 for D >= 34 since 4(D-2) >= 128 > 25) with exact integer certificates below; the log depth is necessary, every fixed K dying at D = 16, 66, 316 for K = 1, 2, 3; the criterion holds directly at every even D = 34..600 and at every depth transition to D = 10^6; the death law D = 2 ceil(5^(K+1)/4) + 2 is known at three depths only; with the odd-D theorem, Conjecture S at base 5 is settled everywhere except strictness at odd D == 1 mod 5, exact through D = 80. Witness: slice-sign-even-half.
• [Verified] The even half at base 3 holds per dimension for every even D = 2..102 and on the grid 106, 110, ..., 178 - rho_D < fill/3 with strictness at each, by the exact-integer Collatz-Wielandt certificate 3 beta_(K+1)(c) < fill beta_K(c), no determinant lemma and no exceptional class D == 4 mod 6 needed; beta_K(0) = b(K) exactly, so K_min >= L*(D) + 1 with L* the last level with V(L) < 0, and K_min = L* + 1 or + 2 at every tested D; spot depths K_min = 8, 50, 140, 291 at D = 12, 30, 50, 72; two implementations sharing no code reproduce all 36 rows to every digit, including the two non-monotone slack rows. Witness: slice-sign-even-half.
• [Verified] The odd half's last gap narrows to the same 2-adic bound: at base 3 and odd D == 1 mod 3, where the mod-3 strictness lemma is silent, v_2(det M_even) < v_2(fill) = (D-1) + v_2(D+2) at every D = 13, 19, ..., 241, silent only at D = 7 (v_2(det) = 7 >= 6, closed by the direct computation det(fill I - 3M) != 0), so rho_D != fill/3 on 13 <= D <= 241 and, with rho_D >= fill/3 at every odd D, rho_D > fill/3 strictly at every odd D <= 241; the reference v_2 rows on the two other classes read 1, 2, 3, 1, 4 (base 3, D == 4 mod 6) and 2, 1, 2, 3, 2 (base 5, even); both exceptional classes of Conjecture S reduce to one uniform statement, an upper bound on v_2(det M_even), and v_2(det M_even) <= n = (D+1)/2 at every D = 13..241 in the class, failing only at D = 7, is exactly strong enough. Witness: slice-sign-even-half.
• [Proved] The even half of Conjecture S at base 3 is a theorem entire: rho_D < fill/3 for every even D >= 2, hence dim_slice < d - 1 at every even D, base 3, middle-digit design, strictness included - by the Fourier/telescoping port to q = 3 (the exact phase cancellation load-bearing, off-centre variants failing with integer witnesses), four nested frequency tracks (+-1 at angle 0, the half-points +-(3^(K+1)-1)/2 at the tripling fixed point pi; on-track prefixes nest, exits never return), exit-cost and window/subtree lemmas giving E_K(D) <= [4(K-1)(0.7528157^(D-1) + 0.7052518^(D-1)) + 2 * 0.2266816^(D-1)] exp(2(K+1) 0.8900159^(D-1)) for everything off the two leader pairs, and the criterion closing at K_1(D) = K*(D) + O(1), analytic for even D >= 38 (182 interval-certified inequalities to D = 400, monotone domination beyond, the margin term the true cosine deficit delta(D) = O(3^(-2 K_1)), < 4.1e-76 at D = 38, since an absolute 1e-8 term fails at D = 399999998), exact certificates below; checks: Fourier form to 9.4e-61, telescoping to 2.6e-59, tracks exhaustive over all 3 !| n < 3^10, the subtree bound never exceeded (worst sigma_5 = 1.023 against 8.97), the E-bound dominating exact enumeration at all 35 (D, K) points and at 28 fresh ones (D in {10, 14, 22, 26} x K in {3..9}, worst ratio 19.74), class counts exact at K = 9, an exit-level sweep to j = 60, D = 4000 finding the caps asymptotically exact (worst attainment 0.999121) but never breached, seven certified constants re-derived to 22 digits by exact interval arithmetic on a 10^-90 grid with outward rounding (a hand-rounded 0.6696 reads 0.66966), and the theorem machine-checked in exact integers at D = 38, 40, 42 (the certificate holds at exactly K_1, fails at 0.8 K*, K_1 = K_min + 1 at all three); the chain's single global safety factor is 2 and the h-exit(2) attainment (two of four residues reach C_H) is load-bearing. Witness: slice-sign-even-half.
• [Proved] The base-3 transient is identified in closed form: L*(D) is the greatest odd integer <= K*(D), K*(D) = [(D-1) ln R + s_D]/ln((D+2)/(D-2)), R = prod_(i>=2) cos(pi/3^i)/cos(2 pi/3^i) = 1.2553249438... - the half-point frequency rides the pi fixed point with per-level magnitude advantage cos(pi/3^i)/cos(2 pi/3^i) > 1 against per-level amplitude cost about (D-2)/(D+2), its sign alternates as (-1)^K (that is the odd-L dip), and the crossing is the transient - so the certificate depth constant is ln(R)/4 = 0.0568486146... and K_min in [L* + 1, ceil(K*) + 2] for even D >= 38; exact on 58 of 58 L* rows, every even D = 6..120, each one exhausted by proof and not by margin - M >= 0 and u_L = M^L e_0 >= 0, so a single t with (M^T)^t v >= 0 entrywise forces V(L) >= 0 at every L >= t and no census window can truncate the answer - with towers L* = 79, 97, 107 at D = 38, 42, 44 and L* = 811 at D = 120; a census carried only to a 4D + c window is unsound past D about 70 since L* is quadratic, but no row of D = 6..120 is in fact false; the column-sum identity fill - 3 colsum(c) = (D-1) v_c, which is prop:mass by root-of-unity filtering and needs no per-row check, makes that row certificate the even-half Collatz-Wielandt test itself, so the stopping level is K_min exactly, L* + 1 on 36 rows and L* + 2 on 22; scoped to D >= 6 since D = 4 has no dip; at D = 10, 20 the half-pair carries the largest magnitude in the spectrum, the dominant pair only third. Witness: slice-sign-even-half, lab/base3-transient-exhaustion.
• [Proved] Base 3 is the unique hard base: the half-point track exists at strength |A_q(-1)|/A_q(1) per level with A_q(-1) = 1 - (-1)^((q-1)/2), so the ratio is 1 exactly at q = 3, 0 at every q == 1 mod 4 (the symbol dies at pi, the base-5 case) and 2/(q-1) < 1 at every q == 3 mod 4, q >= 7. Witness: slice-sign-even-half.
• [Verified] Exact Collatz-Wielandt certificates give rho_D < fill/9 at every even D = 2..56 and rho_D < fill/11 at every even D = 2..74 (machine-pinned to D <= 42 and D <= 60), K_min <= 2, V(L) > 0 everywhere, no transient, and base 9 = 3^2 inherits nothing from base 3; at q = 7 the K = 2 -> 3 step lands at exactly D = 174 as the frontier-race law predicts, the frontier f_2 = 85 converged from D = 160, the asymptotic death law landing there too; earliness (asymptotic death minus true death) is monotone down in K and up in q - the depth-0 death is D = 4 at every base, so base 5 is one even step early at K = 0 (4 against 6) and exact at K = 1, 2, K_0(5) = 1; q = 7: 1, 1, 0 steps; q = 9: 2, 1; q = 11: 2, 2 - the asymptotic law being exact for all K >= K_0(q); at q = 9 the window edge is immune when h == 0 mod q (D = 20, 38), so tightness must be stated mod q; the 12 printed constants of the base-3 chain are asserted against interval endpoints and printed by ceiling. Witness: slice-sign-even-half.

The tent rank law

• [Proved] The palindromic module reformulation: mod 2 at odd D (R = (D-1)/2, n = R+1, M = 6R+2) the symbol is G = (1+t)^(4R)(1+t+t^2) with 4-block coefficients C(R, a), and nullity_2(M_even) = dim{H : no exponent == 1 mod 3, deg H <= M, G | H, t^M H(1/t) = H} - palindromy folds the R+1 kernel conditions onto one residue class mod 3; the explicit kernel vectors H_(b,i) = t^s (1+t^3)^i (1+t)^(2^b) with s = (M - 3i - 2^b)/2, i even, i + 2^b >= 4R, 3i + 2^b <= M are independent since 3R - 1 <= 2^b <= 6R - 4 forces a unique b, and number tent(D), so nullity_2 >= tent(D) with troughs exactly where 3R is adjacent to a power of 2; the reversal involution on the t^3-chain of the single generator gives nullity_even = ceil(nullity_full/2) exactly; the staircase submatrix (rows c' = R - j, leftmost pivots at R - 1 - 3j) yields only floor((R-1)/3) + 1 independent rows, the wrong third of the rank; without the (1+t+t^2) hypothesis the valuation lemma yields only tent + 1 (witnesses R = 2, 4, 7); matrix nullity equals tent at 21 D through 511 and at every odd D = 3..401, the module identities to R = 1024. Witness: slice-sign-even-half.
• [Proved] Lemma M, sharp with equality: for every d >= 2 and r in {0, 1, 2}, the maximal (1+t)-valuation over nonzero H in F_2[t] with no exponent == r mod 3 and deg H <= d is mu_r(d) = max_(2^b <= d) [floor((d - s_0 - 2^b)/3) + 2^b], s_0 = (r + 2^b) mod 3, attained by t^(s_0)(1+t^3)^i(1+t)^(2^b) - the Frobenius split H = A^2 + t B^2 gives the exact case law v(H) = 2v(A) / 2v(B) / 2 min / 2w + 1 (the equal-valuation case forced by C' = B_1^2, a unit at 1), classes move r -> (2r, 2r + 1), the recursion M_r(d) <= Phi(M_(2r)(floor(d/2)), M_(2r+1)(floor((d-1)/2))) with Phi(X, Y) = max(2X, 2Y, 2 min(X, Y) + 1) has the closed form as supersolution by lifting the child's maximising Frobenius block b -> b + 1 (six integer inequalities, X = Y forcing b_e = b_o by a numerator gap >= 2^(min+1) - 3, finite windows d = 5..12 with 24 evaluations, 8 tight, and base cases d = 2..4); the corollary Lemma M' for (1+t+t^2) | H is M'_r(d) = M_r(d - 3) + 1 for d >= 5, the cheapest purchase of valuation being one Frobenius block plus (1+t^3) padding at exchange rate 3:1, which is where sup nullity/n = 1/3 comes from; brute-forced to d <= 16000 (failure set exactly the four d < 2 pairs), the upper bound certified independently by full rank of Lucas submask matrices at d = 1023..8193, the supersolution tight at 4926 points up to 2^60 with minimum slack 0, the case law exact on all H < 2^17, and the true minimum of the module Y computed at 204 R up to 1025. Witness: slice-sign-even-half.
• [Proved] The one-class window lemma, by the parity of an index: Y = Z ∩ G F_2[t] is F_2[t^3]-free of rank 2 with generator degrees delta_1 < delta_2 in distinct classes mod 3 and delta_1 + delta_2 = 12R + 5 exactly - truncation counting gives dim_(F_2) Z/Y = (delta_1 + delta_2 - 2)/3, the projection onto the missing exponent class identifies the cokernel of Z -> F_2[t]/(G) with F_2[u]/gcd(A_0, A_1, u A_2) of dimension exactly 1 (since (1+t^3) | G but (1+t^3)^2 does not), so dim Z/Y = deg G - 1 = 4R + 1; the sum is odd, so delta_1 <= 6R + 2 = M < delta_2 in two lines, margins 0 and 2 impossible and margin 1 iff delta_1 = M; explicitly {delta_1, delta_2} = {12R - 2A + 2[a even], 2A + 3 + 2[a odd]} with a = floor(log_2(4R - 1)), A = 2^a, from A + 2 <= 4R <= 2A; hence nullity_2(M_full)(D) = floor((M - delta_1)/3) + 1 and nullity_2(M_even)(D) = ceil(nullity_full/2) in closed form for every odd D; the identity generalises as delta_1 + delta_2 = 3(deg G - deg_u gcd) + 2 at 400 random G, the closed form holds to R = 200000, margins below 5 lie in {1, 3, 4} exactly as parity predicts, R = 683 = J(11) has delta_1 = M (margin 1) and R = 1365 = J(12) margin 3. Witness: slice-sign-even-half.
• [Proved] The Jacobsthal tent rank law, entire: for every odd D = 2R + 1 >= 3, base 3, middle-digit design, nullity_2(M_even)(D) = tent(D) = 1 + dist(R, {J(a), J(a) + 1}) with a = floor(log_2(4R - 1)), and sharply nullity_2(M_even) <= ceil(n/3), n = (D+1)/2, with equality exactly at D in {3} ∪ {2^(2j) + 1}; with m = R - J(a) all four (parity of a) x (branch) cells reduce to nullity_full = 1 - 2m (m <= 0) or 2m (m >= 1), the parity of a cancelling completely, and halving gives nullity_even = 1 + d_a(R); the nearest-trough index is a itself (an a - 1 reading was rejected exhaustively), margins exactly 1 at both window endpoints propagate by 1-Lipschitzness, R = 1 is the sole reason the cap is ceil rather than floor, and the odd-a peaks miss by exactly 1; closed form equals tent equals the real transfer-matrix nullity at every odd D = 3..1401, closed-form checks to R = 500000 with points to 2^60 (argmin strictly unique everywhere), no residue family past the cap (max excess 0); so v_2(det M_even) = tent(D) + X(D) with the tent capped at ceil(n/3), and base-3 strictness rides on the cascade layers X(D) alone. Witness: slice-sign-even-half.

The 2-adic Smith cascade

• [Verified] The Smith layers are a Jacobsthal cascade and the excess anticorrelates with the tent: full 2-adic profiles at every odd D = 5..511 (254 rows) - octave maxima of L_2 = #{a_i >= 2} are exactly the Jacobsthal numbers J(k-2) (about n/6) and of a_max exactly floor(log_2 D) + 4 on octaves 3 to 8, octave 2 (D = 5, 7) reading L_2 = 1 against J(0) = 0 and a_max = 7 against 6, L_4 <= 1 and every non-spike divisor has a_i <= 3 (L_5 = 1 at 233 of 254 rows, the spike alone), and the min-of-cones consequences of per-layer tents (interior local minima 1, 1-Lipschitz in steps of 2, no plateaus off 1) hold at layers 1, 2, 3; the excess X = v_2 - nullity grows linearly (octave maxima 6, 6, 7, 8, 10, 17, 28, driven by L_2) but peaks at the tent troughs, so the sum stays small: v_2 <= ceil(n/3) + 9 at every odd D = 5..511 (per-octave slack 5, 5, 5, 7, 7, 9, 9, growing like log_2 D, extremal D = 255, 257, 511), v_2 <= n at every odd D >= 9 (equality only at 9, 15, the only violations D = 5, 7), and in the class D == 1 mod 6 (84 rows, 13..511) v_2 <= n - 3 < D - 1 everywhere (max ratio v_2/(D-1) = 7/18 at D = 19 only, 1/3 at D = 13), hence rho_D > fill/3 strictly at every odd D <= 511; the reading "v_2 <= ceil(n/3) + 13 fails at D = 511" is arithmetically false (95 < 99), what died is the tent-plus-excess split, the object to bound being the sum; growth laws beyond D = 511 are unproved. Witness: lab/smith-cascade.
• [Verified] The layer-2 window law: the second 2-adic Smith layer of M_even (base 3, odd D = 2R+1) is a divisor-plus-ceiling window on the kernel family - with H_i = x^s(1+x^3)^i(1+x)^(2^b), b = ceil(log_2(3R-1)), and kernel elements as coefficient polynomials c(z), the mod-2 kernel vectors that lift mod 4 form V_2 = {c : g_D | c, deg c <= C_D}, L_2 = C_D - deg g_D + 1, with generator g_D = z^m c_t(z^(2^e)) (c_t the F_2 Fibonacci polynomials), N = 2t+1 = J(k) Jacobsthal, k + e = b - 1; via the dictionary u^t c_t((1+u)^2/u) = 1 + u + ... + u^(2t) the x-side generator block is the odd-length repunit Rep(N, 2^e)(x^3) of zero (1+x)-valuation, which is why a pure valuation threshold (31/49) and a one-sided ideal (41/49) both fail; Law E gives g = |2R - 2^(b-1) - 1|, e = min{e >= 1 : J(e) >= (g+1)/2}, m = max(0, 2w - c(e)), L_2 = min(w+1, c(e)+1-w) with w = C_D - t 2^e, c(e) = 2J(e-2) - 1, deriving the L_2 min-of-cones tent of height J(e-2) with octave peaks J(b-4); the explicit element H_2 = x^s(1+x^3)^(i_0+2m)(1+x^(2^b))Rep(N,2^e)(x^3) is derived from the Frobenius identity psi^(2^e) = (1+u^(2^e))^2/u^(2^e) and lifts mod 4 at every row; the mod-4 symbol is P == [(1+t^4)^R + 2Rt^2(1+t^4)^(R-1)](1+Dt+t^2); C_D = K at 174/199 rows with deficits in 2J({2..5}) constant per (b,e) slot, two trial ceiling laws failing at D = 249 and b = 9; 199/199 at odd D = 5..401 and 60/60 at D = 403..521, with N = 43 = J(7) appearing at D = 257, 259 and N = 85 = J(8) at the b = 10 peak D = 513; a peak staircase breaks at D = 237 (nontrivial Rep(3,32) at b = 9, invisible below by the J(1) = J(2) = 1 collapse), the block-size identity 3J(k) = 2^k - (-1)^k is a tautology, and the family exponent floor(log_2(4R-1)) is wrong at 57/99 rows. Witness: slice-sign-even-half, smith-window.
• [Verified] det(m_full) = det(m_even) * det(m_odd) exactly, at every base and both parities (the core commutes with carry reflection by palindromy, the symmetric and antisymmetric blocks are the even and odd conventions, conjugation preserves determinants; by Bareiss to D = 101), so v_2(det m_even) <= v_2(det m_full) and the strictness target v_2 < D - 1 can be attacked on the core, whose mod-2 kernel is the one-generator shift module; the core is the coefficient-extraction map E: X -> ([x^(3j+1)](PX)) on deg X <= 2R, in polyphase coordinates the striped Sylvester matrix of (P_1, P_0, yP_2) (exact over Z at D = 5..13), so the window module is a bounded syzygy module, rank-2 free by Hilbert-Burch, and delta_1 + delta_2 = 12R + 5 is the syzygy degree identity, which is why it holds at 400 random symbols. Witness: slice-sign-even-half.
• [Verified] The cascade holds across [512, 2048) with two fresh octaves attained on the nose: max L_2 = J(k-2) at exactly D = 3*2^(k-1) + {1,3} (43 at 769/771, 85 at 1537/1539), max L_3 = J(k-4) (11 at 705/707 and 833/835, 21 at 1409/1411), a_max = floor(log_2 D) + 4 (13 at 1023, 14 at 2047), L_1 = tent(D) at every row through 2047, v_2 <= n everywhere (worst 0.37); at an L_j maximiser the profile is a flat block a_i = j plus one a_max spike, maximiser sites scale D -> 2D-1, and the cascade stacks at tent troughs (L_1 = L_2 at 767/769, L_1 = L_2 = L_3 at 701..707), so X peaks at stacking sites (51 at 767/769), not at the tent troughs (X = 7, 3 at 683/685); at 2^k - 1 sites X = v_2 - L_1 = a_max - 1, while v_2 - ceil(n/3) extends 5,5,5,7,7,9,9,11 as ...,11,11, not ...,11,13 (they part at 2047 where L_1 = 340 < 342, tail [1^339, 14]); hence rho_D > fill/3 strictly at every odd D <= 583 plus 685, 703, 769, 1021; all 42 adjudicated rows agree between two eliminators sharing no code, D = 1409 at precision 512. Witness: slice-sign-even-half.
• [Verified] Base-5 exceptional-class strictness is exact to D = 511: v_2(det M_even) < 2(D-1) <= v_2(fill) at every D == 1 mod 5 - odd class D = 11..511 complete (51 values, v_2 running 2..105 against thresholds 20..1020, smallest margin 143), even class D = 166..506 joining 6..156 (35 values, smallest margin 307) - so rho_D != fill/5 throughout, the range extended from 80; 10 spot rows spanning both classes agree with an independent eliminator on v_2 and full profiles, and the matrix builder agrees entrywise with the graph-search construction at all 10 D. Witness: slice-sign-even-half.
• [Verified] The ceiling law: with Law E's slot data (b = ceil(log_2(3R-1)), g = |2R - 2^(b-1) - 1|, e = min{e >= 1 : J(e) >= (g+1)/2}, k = b-1-e) the ceiling deficit is K - C_D = 2J(e-1) iff k is even, else 0 - k >= 1 for every R (slot-endpoint identity g_max = 2J(b-2) - 1, exact for b = 4..60, no k <= 0 row to R = 60000), so a k >= 2 guard is vacuous, the rows D = 23, 87 once read as k = 0 are e = 3, 5 with k = 1 (deficit 0 by parity), and D = 1367 is b = 11, e = 9, k = 1, deficit 0; e = 2 never occurs; 259/259 at odd D = 5..521 and 643/643 at b = 3..13, D <= 4779 (the full b = 11 octave of 342 rows plus the b = 12, 13 boundary slots), by a Smith-free extraction with no precision parameter; never-seen deficits predicted and attained: 42 = 2J(6) on the whole slot D = 429..471, 10 on 493..503, 2 at 511, 517, 0 at the peak, 86 = 2J(7) on all 84 rows of the b = 11 slot (e,k) = (8,2), 170 = 2J(9) at b = 12, 342 at b = 13; realised deficits {0, 2, 6, 10, 22, 42, 86, 170, 342}, 152 of 212 nonzero-deficit rows nondegenerate (L_2 > 1), the b = 12 peak giving N = 341 = J(10); so L_2(D) is a closed function of R alone through Law E plus the ceiling law. Witness: slice-sign-even-half, smith-window.
• [Verified] The arithmetic amplitude law: max L_j in octave [2^k, 2^(k+1)) equals J(k + 2 - 2j), attained at D = 2J(k+1) + 3 + 2(J(k+2-2j) - 1) with the flat-block-plus-spike profile [j x (J(k+2-2j)-1), spike] - 13/13 at k = 6..10, j = 1..4, including D = 689 (octave 9, L_4 = 3, tail [4,4,6]) and D = 1377 (L_4 = 5 = J(4), block 4x4+7); each octave carries two block towers, amplitudes J(k-2j+2) at the upper trough and J(k-2j+1) at the lower; the k = 10, j = 4 edge D = 1379, 1381, 1383 reads L_4 = 5, 4, 3, so max L_4 = J(4) = 5 sits on a length-2 plateau 1377/1379 and is never exceeded; octave 8 (D = 343..365) gives max L_4 = J(2) = 1 and max L_3 = J(4) = 5 at D = 353; L_1 = tent(D) and v_2 <= n hold at all 31 new rows (worst ratio 0.14). Witness: slice-sign-even-half.
• [Verified] The window-module machinery is classical: the carry core is a generalized (mosaic) Sylvester map of a 1 x 3 polynomial row ("striped Sylvester" is not a term of art), its kernel the truncated first syzygy module, rank-2 freeness is Hilbert-Burch, and delta_1 + delta_2 = 12R + 5 is the mu-basis degree identity mu_1 + mu_2 = n - deg(gcd) (Cox, Sederberg and Chen 1998; the Index Sum Theorem), so neither the identity nor the freeness is claimable and the 400-random-symbol generalisation reproves a 1998 theorem; the one-generator window step is two lines from Forney's predictable-degree property, leaving in-house only the evaluation 12R + 5 for this symbol, which needs the grading stated and polyphase coprimality asserted; claimable after nine recorded empty searches: Lemma M (the closest neighbours bound degrees, never a (1+x)-adic valuation), the Jacobsthal tent rank law (the Jacobsthal literature never uses the sequence as a rank formula's breakpoint set), the 2-adic Smith-layer/window structure (nearest miss: Smith forms over F[y], algebraically closed, no modular treatment), the Bockstein pairing as a layer-2 reader, and F_2 Fibonacci/Dickson kernel generators; three leads open - the full Beckermann-Labahn text, F_2 polyphase filter-bank Bezout twins, and mu-bases in positive characteristic, the last the only plausible threat to Lemma M. Witness: slice-sign-even-half.
• [Proved] Lemma W, the ceiling mechanism: on family coordinates multiplication by z is multiplication by psi = (1+x^3)^2/x^3 = x^(-3) + 2 + x^3 over Z, so the mod-4 obstruction class obeys ob(zc) = Lambda ob(c) mod im(E mod 2) with Lambda = S + S^(-1) folded at the centre (the raw vector identity fails at D = 29; only the class is intertwined); hence if Y_0 corrects the generator (E(Y_0) = obraw(g)) with x-valuation cmin, then psi^i Y_0 corrects z^i g while cmin + 3i <= R, so C - deg g >= min(K - deg g, floor(a_0/3)) with a_0 = R - cmin the maximal correction reach and cmin the corrector's half-support extent from the centre, not a valuation; the uncapped C - deg g >= floor(a_0/3) is false at D = 25 and at 29 of the 115 rows D = 23..251, exactly the cap-strict rows, and equality L_2 - 1 = C - deg g = floor((R - cmin)/3) holds at the other 86, replacing the fitted ceiling by one linear-algebra invariant of the row; escaper independence is equivalent to ceiling exactness, and rank(phi) <= K - C follows from membership alone; the mod-2 family element's degree does not set the ceiling (D = 115: all s_j >= 0 yet C = 3) and a_0 has no affine closed form in C - deg g (D = 47 against 115, a_0 mod 3 varying, the floor load-bearing), so the ceiling law waits on a closed form for a_0 satisfying floor(a_0/3) = K - deg g - 2J(e-1)[k even] plus the single-element membership proof. Witness: slice-sign-even-half.
• [Proved] Lemma S, the symbol reading of the carry core holds at every 2-adic layer: for odd D = 2R + 1, with P = (1 + t^2)^(D-1)(1 + D t + t^2) and H_x = x_0 t^R + sum_(j >= 1) x_j (t^(R+j) + t^(R-j)), the row at c' of M_even x is the coefficient of t^(3 nu + 1) in H_x P at nu = R - c', and H_x P is palindromic about 3R + 1, so the R + 1 rows are exactly the exponent class 1 mod 3 on [0, 6R + 2]; hence for every r >= 1, M_even x == 0 mod 2^r iff H_x P lies in the Z_2[t^3]-module generated by 1, 2^r t and t^2, equivalently, with u = t^3, H = H_0(u) + t H_1(u) + t^2 H_2(u) and P = P_0 + t P_1 + t^2 P_2, iff H_0 P_1 + H_1 P_0 + u H_2 P_2 == 0 mod 2^r; the r = 1 case is the mosaic Sylvester row already recorded as classical and the mod-4 symbol is already recorded, so what is added is the one row (P_1, P_0, u P_2) carrying every layer, checked as sets and not only as dimensions at D = 5..13, r = 1, 2, 3. The layers are not truncated-syzygy dimensions of that row over Z_2[u]: the syzygy module of the row is the kernel of M_full, not of M_even, and the two nullities differ by the already-proved halving nullity_even = ceil(nullity_full/2), because u = t^3 does not preserve palindromy; the operator that does is psi = u + u^(-1), as Lemma W states. Witness: smith-window.
• [Proved] Lemma Lambda, the family shift is intertwined on the nose: write psi = t^3 + 2 + t^(-3), the integer multiplier (1 + t^3)^2/t^3 of Lemma W, and ob(H)(nu) = ((H P)[3 nu + 1] mod 4)/2 on 0/1 palindromic coefficient vectors of the mod-2 kernel; then (psi H P)[3 nu + 1] = (H P)[3 nu - 2] + 2 (H P)[3 nu + 1] + (H P)[3 nu + 4] and (H P)[3 nu + 1] is even, being a mod-2 kernel row, so its doubled term dies mod 4 and ob(psi H) = Lambda ob(H) holds as raw vectors with Lambda = S + S^(-1) folded by nu <-> 2R - nu; the family satisfies H^(j+1) = psi H^(j) - 2 Z_j with Z_j = H^(j) + (t^3 H^(j) AND t^(-3) H^(j)), and Z_j is palindromic and inside the coefficient box because deg H^(K) <= 2R (from i <= (6R + 2 - 2^b)/3) and the overlap sits in [val + 3, deg - 3], so ob(X_(j+1)) = Lambda ob(X_j) + A(Z_j) with A(Z_j) in the image of the mod-2 symbol and the class identity of Lemma W holds at every row with its raw defect named; the pair moves together, and psi = t^3 + t^(-3) with Z_j = H^(j) + AND makes the lift identity false at D = 29, 31, 47, 115, 251. Witness: smith-window.
• [Verified] The layer-2 window is a window, and its generator and ceiling regenerate from the symbol: V_2 = g_D F_2[z]_(<= C_D - deg g_D), g_D = z^m c_t(z^(2^e)) with c_t the F_2 Fibonacci polynomials c_0 = 1, c_1 = 1 + y, c_t = y c_(t-1) + c_(t-2), and C_D = K - 2J(e-1) at even k, K at odd k, at 199/199 rows of odd D = 5..401 and 100/100 of odd D = 403..601; the slot, the window bounds and the closed forms are the shelf lane's arithmetic line for line and only the object side is independent - the kernel family, the mod-4 symbol, the obstruction and the extraction of V_2 - so what this adds is a committed generator for g_D and C_D, which the lane's own scripts do not compute, pinning L_1 and L_2 alone. The ceiling is a corrector length: C_D - deg g_D = min(K - deg g_D, floor(reach/3)) with reach = R - jmax, jmax the least index whose mod-2 symbol columns span the generator's obstruction, and reach itself Lemma W's a_0; the min was chosen after the 5..401 overshoot, so honest support is the 100 fresh rows 403..601, where the floor binds strictly at 60, the cap at 38 and they tie at 2, against 108, 99, 92 over all 299 rows, and the floor-strict rows are exactly the C_D < K rows, both ways. Remark: taking the corrector out of the coefficient box leaves an image of corank exactly 1 in F_2^(R+1) at 129/129 rows of odd D = 5..261, every family obstruction meeting it, so the unboxed layer-2 window is the whole mod-2 kernel and an argument living in the untruncated module cannot see g_D or C_D. Witness: smith-window.
• [Verified] The reach law, the last unknown of Law E's ceiling: with Law E's slot data (b = ceil(log_2(3R-1)), g = abs(2R - 2^(b-1) - 1), e = min{e >= 1 : J(e) >= (g+1)/2}, k = b - 1 - e) give the slot its length N = J(e) - J(e-1), which is 2J(e-2) at e >= 3 and 1 at e = 1, its offset u = (g+1)/2 - J(e-1) - 1 and its position p = u above the octave centre R = 2^(b-2) and p = N - 1 - u below it; then the tent identity min(p, N - 1 - p) = C_D - deg g_D says Law E's window length is the distance to the nearer end of the slot in the slot's own coordinate, and the reach law says reach = R - jmax = 3 min(p, N - 1 - p) + 2 [e even] + [k odd](1 + (p mod 2)), with p == R mod 2 whenever e >= 3 so the parity term is the parity of R; exactly one row per odd octave escapes, the e = 1 row above centre D = 4^m + 3, where reach = 5 for m >= 2 and reach = 3 at D = 7. Off those escaping rows floor(reach/3) = C_D - deg g_D + [k odd and e even], and on them it reads 1 against C_D - deg g_D = 0 with the cap K - deg g_D = 0 as well, so min(K - deg g_D, floor(reach/3)) = C_D - deg g_D at every row: the corrector law's statement carries no span test and its branch is a slot statistic - the floor binds strictly iff k is even and e >= 2, the cap iff k is odd with e even or D = 4^m + 3, and they tie otherwise - reproducing the recorded censuses in floor, cap, tie order as 48, 61, 90 at odd D = 5..401 and 60, 38, 2 at 403..601 with no mismatch, and 2399/2399 to D = 4801. This repairs Lemma W rather than resting on it: the landed uncapped inequality C_D - deg g_D >= floor(a_0/3) is false at D = 25 (K = C_D = deg g_D = 0, jmax = 9, reach = 3, so 0 >= 1) and at 29 of the 115 rows D = 23..251, exactly the cap-strict rows, while the family-capped psi-orbit bound C_D - deg g_D >= min(K - deg g_D, floor(reach/3)) holds throughout, and Lemma W's a_0 is reach and not jmax, since C_D - deg g_D = floor(reach/3) at 86 of those 115 rows and = floor(jmax/3) at none. So only the >= half of the ceiling law is promoted, to a consequence of the Verified reach law and the Verified generator law and not to a proof; the deduction is not span-test-free, since reach is defined by the span test and the psi-orbit needs its corrector valuation maximal; and one half stays open, that z^(C_D - deg g_D + 1) g_D does not lift. jmax is therefore not a 2-adic valuation statistic of R but a slot-tent statistic, and the two rank readings that would replace the span test are Refuted with witnesses, corank(E boxed) = K - C_D + 1 failing at D = 15 and the first dependent column index 2J(e) failing at D = 7. Fit rows are the 399 rows D = 5..801, read once and unadjusted; out of sample are the 800 rows D = 803..2401 swept cold plus D = 4099 and D = 16387, all clean, and the swept ladder covers every class of R mod 8. Witness: smith-window.

Odd-side fills

• [Proved] At odd side n = 2k - 1 the residue split of an axis has k low positions and k - 1 high, so a base-2 flat design fills sum over its corners of k^(zeros) (k - 1)^(ones), and the six designs of the plane read as the polygonal numbers in k: low corner k^2 (A000290), tree k(2k - 1) hexagonal (A000384), carpet k(3k - 2) octagonal (A000567), void 2k^2 - 2k + 1 centered square (A001844), corner and centre 3k^2 - 3k + 1 centered hexagonal (A003215), solid (2k - 1)^2 odd squares (A016754); two_census at sides 3 to 11 returns 8, 21, 40, 65, 96 for the carpet and 6, 15, 28, 45, 66 for the tree. Witness: mrlymath::formulas::counting fill polynomial, mrlyweb two_census, A000567, A000384.

Exposure

• [Proved] The exposed faces of a design's level power obey V(L + 1) = occ V(L) - 2 sum over the axes of P S^L, with occ the tile's filled cells, P its adjacent filled pairs along the axis and S the cross positions whose two end cells are both filled, because two adjacent blocks bury one face per spanning position and the spanning positions multiply by S a level; so V(L) is a sum of the powers occ^L and S^L in every dimension, the carpet perimeter closes as (4*8^L + 16*3^L)/5 (A381517) and the sponge surface as 2*20^L + 4*8^L (A332705), and all four counts fold from the residue corners without rendering the tile. Witness: mrlymath::formulas::surface prediction_matches_census_on_every_cube_code and the_corners_fold_what_the_tile_shows, sequences.

Spin

• [Proved] The average of a picture over the q rotations by 2 pi / q keeps exactly the circular harmonics of order divisible by q, the average over all rotations keeps order zero only, and a design of rotation order g shows lcm(q, g) petals under a screen that turns it p/q of a turn per frame. Witness: mrlynum::spin the_harmonics_read_the_rotation_order, spin.
• [Proved] The rings of a spun square-lattice picture sit at sqrt(n) for n a sum of two squares with weight r2(n) = 4 (d1 - d3), silent exactly where a prime 3 (mod 4) divides n to an odd power, and their Dirichlet series is 4 zeta(s) L(s, chi_4); the hexagonal rings carry 6 zeta(s) L(s, chi_-3); the mass of a spun lattice is the Gauss circle count and Hardy's Bessel series for its error is the ring expansion. Witness: A001481, A004018, A003136, A004016, Hardy 1915, spin.
• [Proved] The exact ring profile of a raster integrates to its fill, int 2 pi r F(r) dr = fill, 512.0 at level 3 of the carpet; the carpet's profile is zero to side/6. Witness: mrlynum::spin the_mass_of_the_profile_is_the_fill, mrlyweb fixture.
• [Proved] The spin mass about the fixed point p_d = d/(q-1) of a filled digit d obeys M(r/q) = M(r)/k exactly, since S(x) = (x+d)/q carries the design onto its d piece and divides the self-similar measure by the fill, so M(r) = r^D G(log_q r) with G of period exactly log q - the ripple's period is an identity and not a fit, valid for r/q below the distance from p_d to the other filled cells, that is r <= side at the corner digit and r <= side/2 at the centre. Witness: mrlynum::spin::mass_within, the_spin_mass_scales_by_the_fill_about_a_filled_corner.
• [Verified] The spin dimension read about the corner fixed point at level 6 over the window 27 <= r <= 729, three whole periods of log 3, gives slopes 1.465054, 1.649432, 1.783588, 1.761814, 1.897854, 1.879522, 2.000100 for codes 79, 95, 127, 239, 255, 495, 511 against the exact log(fill)/log 3, every gap at or below 1.9e-2 and the discretisation of the identity, max |M(3r)/(k M(r)) - 1|, running 5.3e-3 to 1.8e-2; the exact integer shell histogram and the crate profile integral agree to 1.7e-6 on the total and 0.5% at partial radii. Witness: lab/spin-census.
• [Verified] The corner ripple separates both equal-dimension pairs of the census where every density reading is identical: at level 7, 127 against 239 gives ripple gap 0.11984 on drift bar 0.04126 and 255 against the carpet 495 gives 0.12042 on bar 0.01461, with the solid square as the rippleless control at swing 0.00277 under its own bar 0.00578 and a code against its mirror at gap 0.00e0. Witness: lab/spin-census.
• [Verified] The spin spectrum P_m, m = 0..12, read at levels 1 and 2 over all 511 nonempty base-3 plane codes, splits them into exactly 101 spectra, the number of nonempty orbits of the square group, with no pair outside one orbit agreeing to 1e-9: within the family it is a complete invariant of the dihedral class and no spin-isospectral witness exists. Witness: lab/spin-census.
• [Verified] The ring-averaged powder of a design is not Porod: every sliding three-period window slope, over every fractal code at level 7 and at both pad 4096 and pad 8192, stays above -2.28 and so at least 0.72 from the -3 of a sharp interface, while the solid square control slides from -2.75781 to -2.35759, within 0.25 of -3 and never near its own -D = -2. Witness: lab/spin-census.
• [Proved] The level-L Menger sponge blocks every lattice line down its space diagonal that meets its bounding cube: the shadow obeys S_(L+1) = union_d (3 S_L + proj d), and along (1,1,1) the 27 cube digits and the 20 sponge digits project onto the same 19 classes, so the induction gives equality at every level, the count 3^(2L+1) - 3^(L+1) + 1 = 19, 217, 2107, 19441, 176419. Witness: lab/spin-census, A220978, A003215.
• [Proved] The sponge's axis shadow is exactly the Sierpinski carpet, 8^L against the cube's 9^L, dimension log 8 / log 3 = 1.892789: the 20 sponge digits project along an axis onto the 8 carpet digits, disjoint modulo 3. Witness: lab/spin-census.
• [Verified] No direction other than the axis is deficient in the searched window - over the 13 directions with 0 <= a <= b <= c <= 3, read to level 4 against the cube, the axis is the only share that falls with the level, every other rising, (1,1,2) to 0.98568 and (0,1,2) to 0.97090 at L = 4. Witness: lab/spin-census.
• [Proved] A radius sqrt(k)/n of the spun scale-n square lattice, read inside the disc of radius sqrt 2, is new at n exactly when no prime p | n has p^2 | k - the sum-of-two-squares condition at the smaller scale is automatic by a parity argument, so only integrality binds - and hence new(n) = sum_(d | rad n) mu(d) B(2n^2/d^2) with B the counting function of A001481, giving 2, 3, 9, 11, 22, 18, 40, 38, 55, 52, 91, 64, 123, 97, 128, 126, 199, 136, 243, 180, the rule, the identity and a direct union agreeing at every n to 64. Witness: lab/spin-census, A001481.
• [Proved] The Gaussian Farey's local factor is the Jordan totient J_2(n)/n^2 = prod_(p | n) (1 - 1/p^2), the square-lattice analogue of Farey's phi(n)/n, approached from below at rate 1/ln n because B(X) ~ K X / sqrt(ln X): the radical-6 family climbs 0.56250, 0.59813, 0.61126, 0.62594, 0.63276, 0.63801 at n = 6, 12, 24, 48, 96, 192 toward 2/3. Witness: lab/spin-census, A064533.
• [Proved] The spin dimension about the raster centre is undefined for a design with an empty centre digit - the empty digit removes the open square of side side/3 about the centre and hence its inscribed disc, so M(r) = 0 for every r <= side/6 and the centre-spun mass carries neither power law nor ripple over a whole factor of q. The bound is attained, in exact integer arithmetic on doubled coordinates rather than cell centres, which would return hole + 1/2 whatever the hole: the squared distance to the nearest filled cell is (side/3)^2 = 6561 at level 5 for both mrly_d2_b3_239 and the carpet 495, that is side/6 = 40.5 exactly, while 79 empties out to 56.572962. Witness: lab/spin-census.
• [Proved] The spin spectrum reads a pair census and nothing else: for a constant-valued 0/1 render on a raster of side q^L, every P_m is a quadratic form in the cell indicators whose Gram matrix is constant on the orbits of the raster's symmetry group acting on pairs, because turning a pair by theta multiplies both harmonic coefficients by e^(-i m theta) while a mirror at alpha sends c_m to e^(-2 i m alpha) conj(c_m) and the phase cancels in the real part; so P_m is a linear functional of the pair census Phi_L and equal censuses force equal P_m at every order, ring count and truncation, the base-3 plane carrying 11 pair classes at level 1 and 461 at level 2 and the level-1 coefficients solved from 11 independent censuses reproducing mrlynum::spin::harmonics at 1024 rings and m = 0..12 on all 511 codes at worst relative residual 1.14e-14. In dimension 3 the covariant object is the degree-l power summed over its orders, not a single (l, m). Witness: lab/spin-census shape, spin.md the spin spectrum is a quadratic form.
• [Proved] The 101 spectra need level 2: Phi_1 takes exactly 97 values on the 101 nonempty base-3 plane orbits, four pairs lying in distinct square-group orbits with all 11 class counts equal, so each pair's level-1 spectrum coincides identically at every order and resolution; reading P_m at level 1 alone and bucketing greedily at 1e-9 returns 97 buckets, with 45-105 at gap 1.30e-16 and level-2 gap 0.151, 61-121 at 1.03e-17 and 0.0689, 78-102 at 6.51e-17 and 0.253, and 94-118 at 1.64e-16 and 0.105. Witness: lab/spin-census shape, spin.md level 1 alone is not complete.
• [Verified] The 13 orders see 9 of the 11 level-1 census directions, and the odd cap of 3 is exact: the half turn rho is itself in the square group and acts trivially on classes, but half-turning one member, tau: {j, k} -> {rho j, k}, is well defined on classes because rho is central in D4, and g_(m, rho j) = (-1)^m g_(m, j) gives Q_m . tau = (-1)^m Q_m; tau fixes 5 of the 11 classes, so the antisymmetric part has dimension (11 - 5)/2 = 3 and no number of odd orders can exceed rank 3, which the six odd orders reach exactly while the seven even orders reach 6 of a possible 8. The level-1 spectrum is strictly coarser than the census it factors through and splits it into the same 97 classes anyway. Witness: lab/spin-census shape, spin.md the 13 orders see 9 of the 11 census directions.
• [Verified] The completeness is not about base 3, as a statement about the census: over all 2^25 base-5 plane codes the level-1 pair census takes 3993511 values on the 4211743 nonempty square-group orbits with 204856 ties over 423088 orbits and largest tie 8, and over all 2^27 base-3 D = 3 codes it takes 1461693 values on the 2852287 nonempty orbits of the order-48 cube group with 757066 ties over 2147660 orbits and largest tie 32, and every tie breaks at level 2, the budget-capped weight window failing to bind and covering every group, all 204856 out to weight 21 and all 757066 out to weight 24 against level-2 censuses of 24805 and 6325 classes, with the canonical counts matching the Burnside averages 4211744 and 2852288 computed from the cycle index in the same pass. Witness: lab/spin-census shape, spin.md the completeness is not about base 3.

The ratio-set power saving

• [Proved] Every occupied direction of the gasket ratio set obeys max(z_1, z_2) > 2 min(z_1, z_2): the top base-3 digit of m(z_1 + z_2) lies in exactly one of the disjoint binaries m z_1, m z_2 and the other is a sum of distinct lower powers, hence at most (3^t - 1)/2, so the slope z_1/w never lies in [1/3, 2/3]; checked against every occupied weight to 8192, every pair to height 120 and every ray at n = 12, and shown sharp and strict by the adversarial pass at minimum ratio 2.0000004 over 14.3 million pairs at level 15, extremal at (3^14, (3^14 - 1)/2). Witness: lab/ratio-set-saving.
• [Proved] The digit-congruence bound and the weight-layer reduction are one bound: z_1 (1 + r) = r w for r = z_1 z_2^(-1) mod 3^k, and r = -1 mod 3 would force 3 | w, so z_1 = r w (1 + r)^(-1) is determined and z_1 -> r is injective for 3^k > w, giving Z(w) <= 2 |R_k| with no failing weight to 8192; beta < 1 from this side would need sigma_k to fall geometrically, exactly what criticality forbids. Witness: lab/ratio-set-saving, lab/occupancy-decay.
• [Proved] Two relaxations of occupancy, both tight enough to keep the exponent: Z(w) <= 2 N_P(1/w) because slopes of denominator w are 1/w apart, and Z(w) <= Zinf(w) because the band is forward-invariant on every integer it contains, so a 3-adic witness suffices; the backward cone of 0 is {z_2 C - z_1 A : (A, C) a gasket pair} intersected with the band, rebuilt by an independent carry-pair dynamic programme with zero mismatches, and Zinf/Z is at most 1.5295 over the eleven weights tested. Witness: lab/ratio-set-saving.
• [Proved] R_k is indexed by the modulus 3^k and R_1 is empty under the hypothesis u > 0, so |R_k| = 1, 3, 9, 23, 63, 168, 457, 1245, 3423, ... starts at k = 2; both studies carry the identical definition and the same offset, which pins the offset the submission candidate needs. Witness: lab/occupancy-decay, lab/ratio-set-saving.
• [Verified] The weight layer read per weight rather than off a running maximum: log Z(w) / log w peaks at 0.7093 at w = 121 and Z(w) / w^(log 2 / log 3) at 1.5975 at w = 1093 over every w <= 8192, with all twenty-four octave argmaxes binary base 3 as the scan prints for itself; on the repunits (3^k - 1)/2 at k = 9, 11, 13 and the shifts 1 + 3^h at h = 7, 9, 11, 13 the exponent holds inside [0.6223, 0.6818] out to w = 1594324 while unstructured neighbours collapse to [0.2861, 0.4272], and the mean forward reach is 0.2249 to 0.2947 times sqrt(w) off the structured families. Witness: lab/ratio-set-saving.
• [Proved] The block rate of the critical band automaton is bracketed by the parity of the depth. Every column sum of every block B(b, j) is Sum_(c = a mod 3) binom(b, c) = (2^b + 2 cos(pi (b - 2a)/3))/3, so its deviation from 2^b/3 takes only two values, -1/3 and 2/3 at even b and -2/3 and 1/3 at odd b; hence lam_b lies in 2^b/3 + [-1/3, 2/3] at even b and in 2^b/3 + [-2/3, 1/3] at odd b, and the two-sided abs(3 lam_b/2^b - 1) <= 2^(1-b) holds at every b. The parity refines which edge is which and not the rate, and the computed excess 3 lam_b - 2^b is positive at every depth reached, so the upper edge is the live one. Witness: lab/band-return-times ladder.
• [Proved] At the horizon n = bk the column transfer is one matrix per residue fixed in k and L(k, bk) = Sum_j w_j B(b, j)^(k - r0(j)) h_j, the head length r0(j) free of k but not equal to 2: it is 1 at every sector below b = 5, at most 2 at b = 5..10 and at most 3 at b = 11..14. So L(k, bk) obeys a constant-coefficient linear recurrence in k and the block rate lam_b is an algebraic integer. Witness: lab/band-return-times ladder, with an independent residue DP reproducing L(k,4k) and L(k,5k) to k = 12 and factoring both characteristic polynomials in exact arithmetic.
• [Verified] The block rates are exact algebraic integers: lam_4 = 6 from (x-1)(x-3)(x-5)(x-6), lam_5 = 3(5 + sqrt 5)/2 from (x-1)(x^2 - 15x + 45), lam_6 = 13 + sqrt 79, lam_8 = (99 + 9 sqrt 65)/2, every lam_b to b = 14 having an exact minimal polynomial that divides the characteristic polynomial with zero remainder, the degree-four-and-up ones at b = 9, 11, 13, 14 irreducible over the rationals by mod-p distinct-degree factorisation; and the block rate is the largest block spectral radius itself, max_j rho(B(b,j)) agreeing with the certified interval to a relative 1e-9 at every b. Witness: lab/band-return-times ladder.

The registry's integers

• [Proved] The registry's written set is finite at every ceiling: a row renders at most 48 terms, so the 18066 rows write at most 48 * 18066 = 867168 integers however far the ceiling is pushed, and the miss density tends to 1. The adversarial read kills the way the bound was first used - 11133 was read as 0.01284 of that cap, a comparison that is vacuous inside the census window because 867168 exceeds the ceiling 100000; the honest saturation is 11133/100000 = 0.11133, and the finiteness is the only claim of the lane untouched by a change of cap. Witness: lab/integer-census.
• [Verified] The census of 1..=100000 over the whole registry: 18066 rows, 7692 closed, 5044 convolved, 2665 side grid and 2665 level grid, each tier matched against an independent derivation from SPACES, ledger::designs and Measure::applies, none unread; stops 5529 ceiling, 6802 cap, 5735 budget, 390 silent; never/once/multiple 41/31/928 at 1000, 3589/765/5646 at 10000, 88867/2897/8236 at 100000, shares written 0.9590, 0.6411, 0.1113; miss density by decade 0, 0, 0.045556, 0.394222, 0.947533; 347308 (row, integer) incidences against 360703 (row, index, integer), so 13395 double counts are refused, and 29144 terms at or below zero are excluded and reported. Two independent refolds of rows.csv by readers sharing no code with the sweep reproduce every one of these numbers with zero mismatches, and add four checks the study did not run - no duplicate key, max |written| = 48 never exceeded, every in-range head term present in written, every ceiling row's head strictly increasing. Witness: lab/integer-census.
• [Verified] The miss set's arithmetic: 269, a prime, is the first missed integer and 1..268 the longest written run; the longest missed run is 447 wide on 95265..95711 with both neighbours written; 100000 is written by 103 rows; the written share on 10000..100000 by greatest prime factor falls 0.5798, 0.1406, 0.0506, 0.0313, 0.0117 over the bands 1..10, 10..100, 100..1000, 1000..10000, 10000..100000; the tail written count by residue mod 12 runs 1175, 440, 145, 194, 715, 176, 420, 224, 531, 358, 229, 116 for a ratio 10.13, and mod 6 1595, 664, 676, 552, 944, 292 for 5.46; primes 750/9592 with only 158 of the 8363 above 10000; cubes 46/46, fourth powers 17/17, fifth 10/10, sixth 6/6, squares 176/316 with the largest written 97969 = 313^2, carried by the single row mrly_bang_d4_28662.voids.side on 4k^4 - 8k^3 + 8k^2 - 4k + 1. Witness: lab/integer-census.
• [Verified] The first missed square is a cap artifact and not arithmetic: row multiplicity at 96^2, 97^2, 98^2, 99^2, 100^2 is 321, 19, 480, 0, 123, and deepening the rendered window to 96 terms writes at least 228 of the 316 squares and moves the first missed square from 9801 to 38809 = 197^2. The adversarial read kills the mechanism first offered for it - that 9801 is odd and so outside the family (2k+2)^2 - because 97969 = 313^2 is odd and written, and because that family was selected by grepping a head prefix out of the study's own rows; what survives is the frontier, printed by the generator over 96..100 rather than read off a chosen family. Witness: lab/integer-census.
• [Verified] The rendered window is measured rather than assumed harmless: the census at 8, 32 and 48 rendered terms writes 5263, 8749 and 11133 integers, so 5870 of the written set arrive only past term 8 and 2384 only past term 32, and the 8-term miss set opens 269, 281, 302, 311 where the 48-term one opens 269, 362, 422, 443. Rebuilding a row's written column from its 8-term head and the pinned stop rule alone, by Newton forward extension, passes 1306 of 1306 ceiling-stopped rows and 1325 of 1333 cap-stopped rows, the 8 failures being exactly the degree-6 detections an 8-term head cannot certify; the 969 budget-stopped rows carry no rendered length in their head and are declared untestable. Extending only the 1325 rebuilt cap rows to 96 terms gives a strict lower bound on a deeper census: at least 11898 written, first miss moved from 269 to 362, longest written run at least 361. The adversarial read kills the first statement of the rebuild check, which reported a pass-set size as a population size and omitted the budget stop kind entirely; the generator now prints the population by stop kind and the failing degrees. Witness: lab/integer-census.
• [Verified] The champions are small perfect powers: 16 at 2858 rows, 9 at 2811, 4 at 2559, 12 at 2303, 36 at 2270, 64 at 2176, 3 at 1951, 6 at 1883, 8 at 1790, 33 at 1777, all twenty of the top twenty below 65 and carrying 39007 of the 347308 incidences, a share 0.1123; the 366 perfect powers of the window carry 58906 incidences, a share 0.1696 against a density 0.003660, 46.34 times their weight. The adversarial read kills the normalisation the enrichment was first printed under - a straddle-coverage ratio whose value at n = 1 is definitional, since a written span containing 1 must start at 1, and which supplies 27.6% of the squares mean - so the study prints unconditional means on 1..1000 instead: 193.42 over all integers, 995.26 over the squares, 920.58 over the perfect powers. Witness: lab/integer-census.
• [Verified] A champion is a property of a measure column, not of a design: euler.side writes 1 in 695 of its 859 rows, peak.side writes 12 in 809 of its 1261, heights.side writes both 9 and 33 in 765 of its 1261, and the eight integers below 100 that heights.side writes most often are 9, 17, 25, 33, 41, 49, 57, 65, every one 1 mod 8, which is what puts 33 = 3 * 11 tenth in a census otherwise made of powers. The adversarial read kills the mechanism first offered - that heights.side rows are the progressions 1 + s(k-1) - since only 840 of the 1261 rows have arithmetic heads and only 5 start at 1, witness mrly_bang_d2_6.heights.side reading 2, 4, 6, 8; the leader set and its 1 mod 8 law stand as printed. Witness: lab/integer-census.
• [Verified] The tiers split the census cleanly: of the 11133 written integers the closed tier covers 7628 with 3983 exclusive, the side grid 6203 with 2603, the level grid 1826 with 541, the convolved tier 792 with 130; above 30000 there are 2174 written integers and the closed tier covers 1853 of them. Witness: lab/integer-census.
• [Verified] Two OEIS collisions, both explained and neither an identification: the ascending champion set opens 2, 3, 4, 6, 7, 8, 9, 12, 14, 15, 16, 18, a window of A100290 and of A336231 and of no other record in a dump of 398817, all three parting at the thirteenth term with 21, 19 and the census's 20; and the written-per-decade run 9, 90, 859 sits inside A209631 alone, which continues 6689 where the census gives 5452. Both searches index every window of the census sequence and walk every record, so they are exhaustive on both sides rather than sampled at offsets. Witness: lab/integer-census, A100290, A336231, A209631.
• [Verified] No recognizable family is systematically missed: 173 records of the dump hold at least ten distinct integers of 1..=100000 and lie wholly inside the miss set, the longest being A361796 at 41 terms, which at a miss density of 0.88867 has probability about 10^-2.1 and is ordinary across 398817 records. Witness: lab/integer-census, A361796.

Crop census

• [Proved] Crop partition and anti-crop complement: classify puts every cell in exactly one of Out, Cut, In, so the keep-cut and strict crops bracket the boundary, and Shape::Anti flips In with Out fixing Cut, so the crop and the anti-crop under the complementary cut rule partition the filled set exactly; read off the definition and asserted both ways on all 118 printed configurations. Witness: mrlymath::shape, lab/crop-counts.
• [Verified] The inscribed sphere never enters the level-1 sponge: census reads cells [0, 26, 1], the one In cell the empty centre and all 20 filled cells Cut, so the keep-cut crop keeps everything and the strict crop nothing; the inscribed octahedron holds no filled sponge cell fully inside through level 2. Witness: mrlymath::shape, lab/crop-counts.
• [Proved] Exact dead zones of the inscribed crops: the carpet crop is empty for r < 1/6 under ball and diamond alike and the sponge diamond crop for r < 1/3, the central holes' inradii; the sponge ball's exact contact radius is sqrt(2)/6 = 0.2357, witnessed by the filled level-3 cell [13/27, 14/27] x [8/27, 9/27] x [8/27, 9/27] whose nearest point to the centre is (1/2, 1/3, 1/3), so the 1/24 sweep reads empty through r = 5/24 and first cuts at r = 6/24. Witness: crop.md, lab/crop-counts.
• [Verified] Saturation and its one failure: the carpet ball crop holds all 4096 filled cells at level 4 from r = 17/24, the first sweep radius past the circumradius sqrt(2)/2, the sponge ball all 8000 at level 3 from r = 7/8, past sqrt(3)/2, and the sponge diamond never saturates in the sweep, reading in = 5356, cut = 1332 of 8000 at r = 1 since the cube's corners sit at L1 distance 3/2. Witness: lab/crop-counts.
• [Verified] The strict inscribed diamond crop of the full side-2m grid holds exactly 2m(m-1) cells. Witness: mrlymath::shape.
• [Proved] A grid-aligned polytope crop is digit counting: walls on multiples of 3^-k keep exactly the cells with coordinates in integer intervals at level k, the count factors along digit positions as in mrlylab::press, and the crop adds nothing. Witness: crop.md.
• [Proved] Over any triadic window r in [R, 3R) the mean of the crossing count C(r) is Theta(R^(d-1)), from C(r) = A(r) - B(r), the step bound 1 <= |x+1| - |x| <= sqrt(D), the exact sandwich B(3R) - A(R) <= W(R) <= 2 (A(3R) - B(R)) and the bracket lemma B <= M <= A, with constants ((m-1)/2) G_min and (m-1) G_max. Witness: lab/circle-crop mean lines, carpet r = 2187..6560 sum 13758140 inside [11019880, 22055720].
• [Proved] C_full(r) = Theta(r^(D-1)): the shell bound above, and below C_full(r) >= (r/sqrt(D-1))^(D-1) from one crossing cell per orthant lattice point of the first D-1 coordinates. Witness: lab/circle-crop corner assert at every radius, band [2.000152, 2.037038] on the carpet at r = 27..6560.
• [Proved] Pointwise C(r) = Theta(r^(d-1)) holds if and only if Phi(r) = C(r) (3^D/m)^L / C_full(r) is bounded above and below, L the least level with r < 3^L. Witness: lab/circle-crop factor lines.
• [Proved] The digit transform route's budget is sum_(a != 0) |phi_L(a)| |S_r(a/3^L)| = O(r^(D-1)), met term by term only if the l^1 mass grows by at most sqrt(3) = 1.7320508076 per triadic step, and the transfer step at the lattice is h(u) = (m + 3^D - 1)/m = 2 at D = 2. Witness: crop.md the transform route, and where it stops.
• [Proved] At D = 2 the crossing shell is exactly 2r + 1 cells at every integer r >= 1, by the telescoping lo_i = hi_(i+1) of the column intervals with hi_(r+1) := 0; the same count at real radius is 2 floor(R) + 1, so the level-j boxes meeting the shell number at most 2 floor(r/3^j) + 1 and one holds at most 2 * 3^j crossing cells. Witness: lab/circle-crop, asserted at every radius of every carpet level to r = 19682.
• [Proved] The fraction p_j(r) of crossing cells whose base-3 digit vector at position j is one the design omits obeys p_j(r) <= 2 * 3^j (2 floor(r/3^(j+1)) + 1)/(2r + 1) < 2/3 + 3^j/r at D = 2, capping every position with 3^j <= r/30 at 0.7 uniformly in r; it does not transfer to Phi, since the sharpest bound the marginals alone support is Frechet-Hoeffding, C >= C_full (1 - sum_j p_j), and sum_j p_j reaches 1.349974 at L = 8 on the carpet and 1.627693 at L = 5 on the sponge. Witness: lab/circle-crop digits and digitrate lines, asserted in exact integers.
• [Verified] The crossing shell's digits are equidistributed away from the top of the scale: carpet window means 0.111086, 0.111086, 0.111068, 0.111063, 0.111141, 0.109478, 0.109295, 0.166786 at r = 2187..6560 against 1/9, the whole departure in the top three positions and locked to L - j, fine positions inside [0.108363, 0.111141] and the scaled drift (p_j - 1/9) 3^k/3^j inside [-0.111806, 0.063806]; sponge 0.259211, 0.259237, 0.259663, 0.256864, 0.286061 against 7/27 with fine positions inside [0.259103, 0.259237], wholly below the null on three readings. Witness: lab/circle-crop digitrate and digittotal lines.
• [Verified] Pairwise digit dependence in the crossing shell is bounded per pair and falls off with the gap: consecutive ratios 1.000165, 1.000219, 1.000521, 0.998543, 0.992613, 1.076698, 0.956158 and gap-two 1.000298, 1.000022, 1.000203, 1.000221, 1.003261, 1.005671 on the carpet at L = 8, over every window [0.939130, 1.714286] and [0.988460, 1.126957], sponge [0.954573, 1.151415] and [0.997413, 1.019127]. Witness: lab/circle-crop digitpair and digittotal lines.

The digit-restricted Mobius meter

• [Proved] Carry-free scaling ties the digit designs' Mobius meters together: for F = a F' inside {0..q-1}, m -> a m is a digit-length-preserving bijection S_F' -> S_F (each scaled digit stays below q, so no carry occurs), giving M_F(q^L) = sum mu(a m); a square factor in a kills the meter identically (F = {0,4} at q = 5: zero at all 21 levels), and prime a = p gives M_(pF')(q^L) = -sum_(p not | m) mu(m), the {0,2} column at q = 3 reading as the {0,1} column twisted by the Thue-Morse sign of the binary index; asserted at every level on all eight scaled census families. Witness: mobius.md, lab/mobius-designs.
• [Proved] The base-4 anti-symmetry M_{0,2}(4^L) = -M_{0,1}(4^L): 4 | q forces every element of S_{0,1} to 0 or 1 mod 4, so even elements carry mu = 0 and M_{0,2}(x) = -M_{0,1}(x/2) at every real x, running maxima included since S_{0,1} is empty strictly between (4^L - 1)/3 and 4^L; exact at all 22 levels, -110/110 at L = 15, 34/-34 and shared Mmax = 1553 at L = 22. Witness: mobius.md, lab/mobius-designs.
• [Proved] No Euler product for a digit design: S_F is not multiplicatively closed, witness 4 = 11_3 and 13 = 111_3 in S_{0,1} at base 3 with 4 x 13 = 52 = 1221_3 outside, so M_F is not the coefficient sum of an inverse Dirichlet series; the series itself is built literature (abscissa Kohler and Spilker 2009, continuation and poles Burnol 2026) and carries no Mobius sum anywhere. Witness: mobius.md, REFS.md.
• [Verified] The digit-restricted Mobius census: exact M_F(q^L) and running maxima max |M_F(x)| for all 38 digit sets with 2 <= k <= q - 1 at q = 3, 4, 5 (depths 24, 22, 14, 21, 13, 11 by class), the ten base-10 one-digit-excluded columns to 10^8, and full-set controls to 3^17, 4^13, 5^11, 10^8; factorization and sieve agree on the q = 3 {1,2} family at every level to L = 16, the base-10 control reproduces A084237, and an independent second-language recompute matched 99 sampled rows exactly. Witness: lab/mobius-designs, mobius.md, A084237.
• [Proved] A power saving for the Mobius meter on the dense digit columns, under GRH: assume L(s, chi) has no zero in sigma > 1/2 for every Dirichlet character chi, let q >= 3690 and let F omit exactly one digit; then for every eps > 0 and all x >= 2, |M_F(x)| <<_{q,eps} x^(3/4 + c_q + eps) with c_q = log PB_q(1)/log q, PB_q(1) = 1 + Phi_q/q and Phi_q = (4/pi) q + (2q/pi) H(ceil((q-2)/2)) + (1 - 2/pi)(q-2) + 0.727, and 3/4 + c_q < alpha_q = log(q-1)/log q, so |M_F(x)| <<_{q,eps} A_F(x)^(1 - delta_q + eps) with delta_q = (alpha_q - 3/4 - c_q)/alpha_q > 0, every fixed delta' < delta_q delivered and the endpoint never; orthogonality mod q^L, the shifted-grid l^1 recursion c_L <= B_q(F) c_{L-1}, the kernel bound B_q(F) <= q PB_q(1) from sin(pi v) <= 4v(1-v) and 1/sin x <= 1/x + 1 - 2/pi with Parseval exact on the excluded digit, and the assembly with its geometric sum are derived, and the uniform max_theta |sum_{n <= x} mu(n) e(n theta)| <<_eps x^(3/4 + eps) of Baker and Harman 1991 is quoted at source; the corollary at m excluded digits runs whenever PB_q(m) < (q-m) q^(-3/4), which holds at m <= 6, 78, 451 at q = 10^4, 10^5, 10^6 and asymptotically for m <= q^(1/2)(1-o(1)), and c_q -> 0 gives delta_q -> 1/4. The attempt to break it drives the chain below the wall, where the failure is quantified rather than hidden (c_q = 0.28087 against alpha_q = 0.999855 at q = 1000), checks the exponent test against the constant-space certificate gap_q(m) = (q-m) q^(-3/4) - PB_q(m) > 0 at every 3 <= q < 20000, the cancellation-reduced and direct forms of delta_q against each other to 10^-9 relative at every printed base, and Phi_q against the exact shifted-grid kernel sum on a 4001-point grid at q = 50, 101, 200, where it is loose by under 20%. Witness: mobius.md a power saving under GRH at large base, lab/mertens-numerology.
• [Proved] The ladder above that theorem, and its floor: for 1/2 <= a < 1, if L(s, chi) has no zero in sigma > a for every Dirichlet character then the same five steps give |M_F(x)| <<_{q,eps} A_F(x)^(1 - delta_q(a) + eps) with delta_q(a) = (alpha_q - b(a) - c_q)/alpha_q > 0 at every q >= q_0(a), b(a) the smaller of the Baker and Harman 1991 table and Zhang 2024 Theorem 1.1 (Zhang strictly smaller inside (1/2, 4/7) and equal at both ends, by the factorisations -5(a - 1/2)(a - 2/5)/(4 - 2a) and -7(a - 4/7)(a - 4/5)/(4 - 2a), with b(a) >= 3/4 throughout), so every common zero-free half plane buys the saving and GRH is only its first rung, the price of a weaker hypothesis being paid entirely in the base; the wall q_0(a) exists and is a true least base at every a, since PB_{q+1}(1) - PB_q(1) < 1.291/(q-2) for q >= 40 while the mass term gains (1-b)(q+1)^(-b) per step, so the gap steps up at every q >= Q(b), the least q with (1-b)(q-2)(q+1)^(-b) >= 1.291, and below that it is negative: exhaustively on 3 <= q < 3690, and on [3690, Q(b)] by a majorant with one interior minimum whose endpoint values are both negative. The attempt to break it looks for a rung the floor misses and finds none: at every b in [3/4, 1), printed rung or not, minimality of Q(b) gives gap_{Q(b)}(b, 1) < -1.56 and a majorant below -0.95 at both ends, with any b < 1417/1850 forcing Q(b) <= 1486 and an empty range, the constants reproduced on a b-grid across the whole interval. Witness: mobius.md a power saving under GRH at large base, lab/mertens-numerology.
• [Verified] The rungs of that ladder: (a, b(a), source, q_0(a), Q(b)) reads (1/2, 3/4, both, 3690, 723), (13/25, 1417/1850, Zhang, 8578, 1486), (11/20, 913/1160, Zhang, 33547, 4754), (4/7, 4/5, both, 92317, 11221), (3/5, 4/5, BH, 92317, 11221), (2/3, 5/6, BH, 3107080, 216023), (3/4, 7/8, BH, 6939524168, 129458304), then (4/5, 9/10, BH, <= 3.09358e13, 128606353005), (9/10, 19/20, BH, <= 3.23663e34, <= 1.73431e28) and (19/20, 39/40, BH, <= 9.24614e83, <= 3.30712e68), a wall printing as an exact integer only below 2^53 with both neighbouring gaps above 1024 ulps and otherwise as an upper bound on the least q; the GRH rung reproduces the wall 3690 and the margin there is delta_q <= -2.395807653 * 10^-6 at q = 3689 against delta_q >= 5.863425182 * 10^-6 at q = 3690, with gap_q(1) <= -1.533059397 * 10^-4 and >= 3.752213034 * 10^-4; the m-budget at q = 10^7 falls 1971, 1002, 365, 176, 176, 8 along the rungs below that base. The attempt to break them reproduces every wall under 4 * 10^6 by an exhaustive scan from q = 3 against the bisection, requires Q(b) < q_0(a) at every rung, sweeps 3 <= q < 3690 for an early close at every rung and finds none, and pins each rendered row as a string. Witness: lab/mertens-numerology, mobius.md a power saving under GRH at large base.
• [Proved] The l^1 floor is a wall on the method, not on the problem: sum_{r mod q} |g_F((t+r)/q)|^2 = q k exactly, so sum_{r mod q} |g_F((t+r)/q)| >= q k / max_r |g_F| >= q for every t, the shifted-grid recursion never contracts, B_q(F) >= q and c_q >= 0 at every base and every digit set; hence the decomposition needs alpha_q > 3/4, that is k > q^(3/4), and every fixed-k column, F = {0,1} at q = 3 included, is beyond it with or without GRH, so it never meets the census or the exponent conjecture. The same floor kills the two neighbouring routes: Davenport's unconditional x (log x)^(-A) in the quoted step exceeds A_F(x) by the power x^(1 - alpha_q), so no unconditional saving follows inside this decomposition without an input of zero-free-strip strength, and Cauchy-Schwarz with Parseval on both factors gives exponent (1 + alpha_q)/2 > alpha_q, worse than trivial. The attempt to break it hunts a negative c_q over 3 <= q < 5000 and a PB_q(1) below 1 and finds neither, Parseval forbidding both. Witness: mobius.md a power saving under GRH at large base, lab/mertens-numerology.
• [Verified] The cost-out of that saving against a hypothetical Type I defect: with the saving delta_q set beside the defect exponent m/(2(q-m) ln q) carried by a level-x^(alpha_q/2) distribution bound for the digit strings, a bound no page here states, the saving is below the defect at the wall (5.86342e-6 against 1.65022e-5 at q = 3690, a factor above 2.8) and above it from q = 3692 on, the least such base in a scan of 3690..10^5 in which the difference rises at all 96310 steps, monotonicity beyond the scan unproved; at q = 10^9 it is 1.16951e-1 against 2.41275e-11, and the tightest corollary row q = 10^6, m = 451 reads 3.14081e-5 against 1.63296e-5. The attempt to break it checks the crossover for a premature crossing at q = 3690, 3691 and for a single down-step in the scan and finds none, and holds the yardsticks apart: delta_q is normalised to the mass, so as a power of x the saving is x^(alpha_q delta_q) with alpha_q >= 0.99993 on every row compared, while the defect multiplies k^L. Witness: lab/mertens-numerology, mobius.md a power saving under GRH at large base.
• [Verified] The coefficient sequence a real Vaughan decomposition hands the bilinear sum is not the sequence that beats the method's diagonal floor: at the eight swept boxes with both sides above x^(2/5), the boxes the identity produces, the Type II coefficient sum_{d | l, d <= x^(2/5)} mu(d) takes values in {-1, 0, 1} at seven of the eight and its full quadratic form sits in [0.6929, 1.2045] of its own diagonal, where a sign vector engineered against the column reads 0.2043 on such a box; over all eighty coefficient cells of the census, sixteen boxes, four families, two depths, two cuts and five real sequences, the form over the diagonal stays in [0.3138, 52.6676] with none below 0.1 and every departure from the swept band upward. Witness: lab/rho-decoupling section menergy signed vaughan.
• [Proved] The large-values refinement of the moment route is the l^2 route itself: splitting the grid at |hat F_L(a/q^L)| >= k^L x^(-eta), bounding the large set by its l^2 mass under the fourth moment and Parseval and the rest by the threshold, all against Parseval on the bilinear side, gives the exponent max(min(alpha + 1/2 - eta, (1 + alpha)/2), min((1 + alpha)/2, alpha + (nu_4 + 2 eta)/2)) = (1 + alpha)/2 identically at every eta >= 0 and every digit set with alpha < 1, and the large-sieve constant of any grid subset for q^L consecutive frequencies is q^L exactly, so no spacing enters. Witness: lab/rho-decoupling section riesz large values chain, 66 rows with c = -(1 - alpha)/2.
• [Verified] The large frequencies are adjacent or isolated grid points, 407 in 331 runs at {0,1} base 3 L = 12 eta = eta_4 against the fourth-moment count 4096, least gap 1/q^L, large-sieve constant on D_L in 1.06009e5..2.13280e5 nearly at its l^2 floor 1.05611e5; at the eight dense census cells the Type II sum at a_m = b_l = 1 on the box M = N = floor(x^(1/2)/2) is the representation count, 0.26 to 0.41 of k^L, and the balanced sum at a_m = 1_(q | m), b_l = 1 a fixed share of k^L, so no bound uniform over bounded coefficients holds there; the sparse cell {0,1} base 100 L = 3 is void at the box. Witness: lab/rho-decoupling sections riesz large values and riesz large values witness.
• [Proved] The second-largest grid value of the digit transform is max_(a != 0) |hat F_L(a/q^L)| = k^(L-1) max_(b != 0 mod q) |g_F(b/q)|, equal to k^(L-1) at one excluded digit and at {0,1} base 3, so the large set is the zero frequency alone exactly below eta_1(L) = log(1/gamma_1)/(L log q), a threshold that vanishes with depth. Witness: lab/rho-decoupling section riesz large values cells, eight cells at 1/k.
• [Proved] The major-arc input for mu on a digit design is effective. Let F be a base-q digit set with k >= 2 every prime of whose digit-difference gcd divides q, condition (E) in one dimension, and let x = q^L. Every real primitive Dirichlet character whose modulus has all its primes dividing q has conductor dividing 8 rad(q), so the possible exceptional zeros run over a set of size bounded in q and Siegel's theorem is never invoked; with that, x^(-1) Sum_{a in M} hat F_L(a/x) S_mu(-a/x) is at most k^L exp(-c sqrt(log x)) with c effectively computable, over the arcs |a/x - b/d| <= (log x)^C/x with d <= (log x)^C, and there is no main term at any arc. Witness: mobius.md The pair route.
• [Proved] Under condition (E) in one dimension and the large sieve Sum_{d <= Q} Sum_{gcd(b,d)=1} |hat F_m(b/d)| << k^m (Q^(2 alpha_1) + Q^2 q^(-m(1 - alpha_1))) at every scale m <= L, with alpha_1 < 1/2 the sup-over-shift l^1 exponent, a digit design's level of distribution survives restriction to an initial segment: Sum_{d <= Q, gcd(d,q)=1} max_{y <= x} |#{n in D_L : n <= y, d | n, gcd(n,q)=1} - (1/d) #{n in D_L : n <= y, gcd(n,q)=1}| << k^L (log x)^(-B) at Q <= x^(1 - alpha_1)(log x)^(-C), the same level as the full-range statement and one power of log x less saving, because the transform's error is uniform in the target residue and the segment splits into at most k blocks per scale. Witness: mobius.md The pair route.
• [Proved] The hybrid bound that carries the digit-restricted bilinear estimate holds at every base with the digit set's own dimension as its exponent. Let F be a base-q digit set with k = abs(F) >= 2, alpha = log_q k, sup-over-shift l^1 exponent alpha_1, and assume the shifted and perturbed large sieve it supplies by Farey spacing, sup over shifts of Sum_{a <= d} sup_{abs(eta) <= delta} F_Y(a/d + shift + eta) << (1 + delta d)(d^(alpha_1) + d Y^(-(1 - alpha_1))) at every scale. For D, E, Y, Q_1 powers of q with D E << Y, Q_2 >= 1, q_1 ~ Q_1 coprime to q and d ~ D all of whose primes divide q, the sum of F_Y(a/(d q_1 q_2) + eta) over q_2 ~ Q_2 coprime to q, over a < d q_1 q_2 coprime to d q_1 q_2, and over abs(eta) <= E/Y with (eta + a/(d q_1 q_2)) Y an integer is << (D E)^(alpha_1) (Q_1 Q_2^2)^(1 - alpha) + E^(alpha_1 + alpha/2) D^(1 + alpha/2) Q_1 Q_2^2 Y^(-alpha/2). Both exponents come from Parseval on a window q^r, where int F^2 = q^(-r alpha) exactly when 0 is in F and ≍ otherwise, so the base-10 values 1/21 and 10/21 are 1 - alpha rounded up and alpha/2 rounded down. Since alpha + alpha_1 >= 1 at every design, this never loses to the plain l^1 bound in the modulus aspect. Witness: mobius.md The pair route.
• [Proved] The lattice half of the digit-restricted bilinear estimate transfers to every base, and the five inequalities it asks are free below 1/3. With x = q^L, the window N K >= x^(1 - 2 beta), delta >= N/x and Q <= x^(1/2), the sum of F_x(a_1/x) F_x(a_2/x) over pairs whose large contribution comes from a rank-2 lattice is << (log x)^5 (Q + E)^(-eps/4) x/(N K), the source's own log power, whenever 2 alpha_1 < alpha, (2 - alpha) 2 beta < 1 - alpha_1, 2 beta (alpha_1 (3 - u) + u - 1) < u alpha/2 for some u in (0, min(1, 2 alpha_1/alpha)], 5 beta < 1 + alpha/2 and 2 beta < (1 - alpha_1)(1 - alpha_1 + alpha/2). The source writes a numerical check for the second and the fourth only; the first, third and fifth are read off steps it performs silently. All five are monotone in the three exponents, so the corner alpha = 1 - alpha_1, beta = 1/4 decides them, and every one holds under alpha_1 < 1/3, beta <= 1/4 and the l^1 floor alpha + alpha_1 >= 1, with 1/3 sharp since three become equalities there. The floor and the threshold on beta alone do not suffice, as alpha_1 = 0.40, alpha = 0.60, beta = 1/4 shows. Base 10 clears all five as published. Witness: mobius.md The pair route.
• [Proved] The pair route's eight inequalities are two. Write alpha = log_q k for a base-q digit set's dimension, alpha_1 for the sup-over-shift l^1 exponent of its transform and beta for the exceptional-set threshold. Of the eight inequalities the route asks, one is a ceiling on alpha_1 alone, 2 alpha_1 < alpha, and seven are caps on beta at fixed (alpha, alpha_1); four of those fall in alpha_1 and two are constant in it, so each takes its minimum over the region at the wall alpha_1 = alpha/2. At that wall the lattice cap (2 - alpha) 2 beta < 1 - alpha_1 and the geometric-mean condition read exactly 1/4, the last lattice cap reads (2 - alpha)/4 and the fourth (1 + alpha/2)/5, all identities in alpha, so none of them ever cuts below the window threshold 1/4 inside the wall. For alpha in (1/2, 1) the region is therefore exactly alpha_1 < alpha/2 and beta <= min(1/4, (2/5)(1 - alpha_1)), the greedy cap cutting from alpha_1 = 3/8 and from nowhere else. A sweep of 66000 cells, 264000 cap tests, finds no exception, and the two wall equalities hold at each of 330 rational alpha. Witness: lab/mobius-region verb boundary.
• [Proved] The exceptional-set threshold obeys the same Parseval floor as the l^1 exponent, and the pair route reaches only k >= q^(3/4). The normalised transform is at most 1 pointwise, so the moment exponent m_t is non-increasing in t; and m_2 = 1 - alpha exactly, since two length-L digit strings congruent modulo q^L are equal. Hence m_t >= 1 - alpha for every t <= 2, and since 2 - t <= 1 for t >= 1 the threshold beta = inf over t in [1,2) of m_t/(2 - t) is at least 1 - alpha at every base and every digit set, the same floor alpha + alpha_1 >= 1 puts on the l^1 exponent. The route's window condition beta <= 1/4 alone then forces alpha >= 3/4, that is k >= q^(3/4), with equality only when the l^1 floor is also an equality. That window condition is a convenience rather than a necessity, and dropping it does not widen the route: on the single-window branch, which carries the greedy step under alpha_1 <= 1 - (13/4) beta, the same two floors give alpha >= 13/17 = 0.764705..706, so that branch reaches only k >= q^(13/17) and the gate rises. The weaker reading k > sqrt(q), which follows from alpha_1 < 1/2 alone, stays true and is simply not sharp, so no earlier row is contradicted. Witness: lab/mobius-region verb check.
• [Verified] The census of the pair criterion over 49 digit designs, and a second machine at base 21. Over the 38 proper digit sets of q = 3, 4, 5, the ten base-10 one-missing-digit columns and base 21 missing 0, one design clears the criterion, 47 are refuted and one is open, the open cell being q = 5 with F = {0,1,3,4}, where the transform vanishes inside a window cell and the infimum matrix loses a row. A pass is decided at the pessimistic corner and a failure at the optimistic one, every cap being monotone in each parameter. A second implementation of the window method returns alpha_1 in [0.2499715, 0.2499822] for base 21 missing 0 at five window digits and sub-scan 8, against the five-digit [0.2499715, 0.2499821] already certified, agreeing on the lower bound to all seven printed digits and differing by one unit in the last on the upper; both run the same method at the same depth, so the agreement witnesses transcription and the upper-bound gap is the only independent information. The same machine reproduces base 10 missing 5 at alpha_1 in [0.3505101, 0.3506471], m_(235/154) <= 0.1362891 and beta <= 0.2875140 against the three published values 27/77, 59/433 and 23/80. Witness: lab/mobius-region verbs criterion and params.
• [Proved] The line half of the digit-restricted bilinear estimate and its two bookkeeping steps, at every base. With x = q^L, a threshold beta admissible and at most 2/5, which with the Parseval floor beta >= 1 - alpha forces alpha >= 3/5 on the design, delta >= N/x, N K >= x^(1 - 2 beta), K above the absolute constant of the pair dichotomy, and N >= x^(eps + max((5/4) beta, (5 beta - 1/2)/3)), the pair sum over the pairs whose large contribution lies on a line is << (log x)^(O(1)) x^(-eps') x/(N K) for x past a point depending on q, k and eps, with eps' a function of eps and the implied constant depending on those three alone. The statement asks nothing of the l^1 exponent and asks of the dimension only what the admissibility of beta already encodes, so the whole l^1 content of the route sits in the lattice half and the greedy step. Two write-outs complete it. For coefficients bounded by the j-fold divisor function, orthogonality on the grid with tau_j^2 <= tau_(j^2) gives #{a mod x : the exponential sum is at least x/C} <<_j C^2 (log x)^(j^2 - 1), so a Heath-Brown decomposition costs a log power where a 1-bounded sequence costs none. And Cauchy-Schwarz in the long variable turns the bilinear sum into x/N times the pair sum of the transform against the sum over l_1, l_2 <= N of min(x/N, the inverse distance from (a_1 l_1 - a_2 l_2)/x to the nearest integer), which is the exact step at which all four coefficient factors leave by the triangle inequality; the dyadic split into level sets and pair-mass classes costs two more log powers. Witness: mobius.md The pair route.

Digit strings across divisors

• [Proved] Orthogonality for digit strings against a divisor: N_F(L; d, r) = (1/d) sum_{a mod d} e(-a r/d) prod_{j < L} g_F(a q^j/d) with g_F(t) = sum_{f in F} e(f t), by expanding the divisibility indicator in additive characters mod d, the digits being independent so the character sum factors over positions, the a = 0 term giving k^L/d; the attempt to break it rebuilds the whole residue vector by dynamic programming against brute-force string enumeration at four bases and checks its total against k^L at every census cell, with no mismatch. Witness: mobius.md digit strings across divisors, lab/rho-decoupling.
• [Proved] The uniform geometric equidistribution bound: for k >= 2, d >= 2, (d, q) = 1 and gcd(d, Delta_F) = 1 with Delta_F the digit-difference gcd, |N_F(L; d, r) - k^L/d| <= ((d-1)/d) k^L (1 - 8/(k^2 d^2))^L <= k^L exp(-8 L/(k^2 d^2)) for every r and L >= 1, since |g_F(a/d)|^2 = k^2 - 4 sum_{f < f'} sin^2(pi a (f' - f)/d) and d | a(f' - f) at every pair would force d/gcd(a, d) | Delta_F hence d | a; the attempt to break it asserts the weaker form as an exact integer inequality at every census cell where the hypotheses hold, five bases and depths to L = 96, with no failure and largest observed-to-bound ratio 0.187 at q = 100, F = {0,1}, L = 16, and the hypothesis edge d = 2, k = 2 holds at bound factor (1 - 1/2)^L; the d^(-2) in the exponent is sharp in shape, since d | q - 1 with F an arithmetic progression of difference m' and a m' = 1 mod d gives |g_F(a/d)|/k = sin(pi k/d)/(k sin(pi/d)) = 1 - Theta(k^2/d^2). Witness: mobius.md digit strings across divisors, lab/rho-decoupling.
• [Proved] The dense-digit bound: for F = {0..q-1} minus E with m = |E|, k = q - m and (d, q) = 1, gamma_F(d) <= (d/2 + m)/k because g_F is the full Dirichlet kernel less g_E, |D_q(a/d)| <= 1/(2||a/d||) <= d/2 and |g_E| <= m, so for d/2 + m < k the error is at most k^L ((d/2 + m)/k)^L uniformly in r; the attempt to break it looks for the gain at fixed digit count, where the bound is vacuous and stays vacuous - at d = 7 the per-digit rate falls 0.4869, 0.3312, 0.2484, 0.1104, 0.0167 as k runs 2, 3, 4, 9, 99 but reads 0.4992 for F = {0,1} at q = 100, against the same ceiling 0.9010 that F = {0,1} carries at q = 3. Witness: mobius.md digit strings across divisors, lab/rho-decoupling.
• [Proved] A power saving over a whole level for dense digit sets: for eps in (0,1), q >= 4^(1/eps), m <= q^(1-eps)/2 and L >= 4/eps, every 2 <= d <= q^(1-eps) coprime to q has per-digit factor (d/2 + m)/k <= q^(-eps/2), so sum over those d of |N_F(L; d) - k^L/d| <= k^L q^(1 - eps L/2) <= k^L x^(-eps/4) at x = q^L, a level of distribution q^(1-eps) with no conditional input; the attempt to break it pushes the level past a constant power of the base and fails, since summing the geometric bound alone caps the level at d ~ sqrt(L)/k, and the census argmax at every family's deepest level is a divisor of q^t - 1 with t <= 8, where no per-factor bound decays. Witness: mobius.md digit strings across divisors, lab/rho-decoupling.
• [Proved] The exact split across the base's own divisors: for d = d1 d2 with d1 | q^m, m <= L, and (d2, q) = 1, the low m digits fix the value mod d1 and reach the rest only through the invertible multiplier q^m mod d2, so N_F(L; d) = sum over w in F^m with d1 | val(w) of N_F(L - m; d2, r_w) with r_w = -val(w) (q^m)^(-1) mod d2, and the density splits as rho_F(d1 d2) = (N_F(m; d1)/k^m)(1/d2); the attempt to break it tests the natural guess 1/d1 for the base part and refutes it, the base part being a digit-string count, with the identity itself pinned against direct enumeration at q = 6, d = 10. Witness: mobius.md digit strings across divisors, lab/rho-decoupling.
• [Proved] The digit-gcd hypothesis is a wall: if gcd(d, Delta_F) > 1 there is no equidistribution, witness q = 3, F = {0,2}, d = 2, where every value is even, N_F(L; 2) = k^L and the normalized error d |N_F(L; d) - k^L/d| / k^L is exactly 1 at every L; the attempt to break the wall by sweeping the whole range instead of one divisor leaves it standing, the unrestricted worst error over d <= 200 reading 1.0483 at L = 32 pinned at d = 164 against 0.019166 once d is required coprime to Delta_F, and such families reduce to a primitive one through the scaling bijection S_(aF') = a S_(F'). Witness: mobius.md digit strings across divisors, lab/rho-decoupling.
• [Proved] The second moment across residue classes: sum_{r mod d} (N_F(L; d, r) - k^L/d)^2 = (1/d) sum_{a not 0 mod d} prod_{j < L} |g_F(a q^j/d)|^2, by Parseval mod d on the orthogonality identity, the mean being the a = 0 term and no cross terms surviving; the attempt to break it looks for a hidden hypothesis and finds none, the identity holding for every d >= 1 and every F, including the walls where the supremum bound is worthless, which is what makes it the one handle left at a pinned divisor. Witness: mobius.md digit strings across divisors, lab/rho-decoupling.
• [Verified] The divisor census of digit strings: exact dynamic-programming counts of N_F(L; d) over 2 <= d <= D for ten families at q = 3, 4, 5, 10, 100, depths to L = 96 and D to 500, printing the worst normalized error, the multiplicative order of q at the argmax, the per-factor ceiling gamma_F(d) and the slack against the proved bound; the counts are pinned against brute-force string enumeration at four bases, the residue vector totals k^L, and the argmax is a pinned divisor of q^t - 1 with t <= 8 at every family's deepest level, d = 164 at q = 3, d = 143 at q = 10, d = 101, 303 at q = 100, with shallow depths straying (d = 199, ord = 99, at q = 10, L = 6). The slow column is the sparse one: F = {0,1} at q = 100 reads worst normalized error 28.593, 14.590, 9.0340, 7.2034 at L = 16, 32, 64, 96, per-digit factor 0.9929. Witness: lab/rho-decoupling, mobius.md digit strings across divisors.
• [Verified] The signed pinned sum against its absolute sum: over the squarefree moduli e = (q^t - 1)/g, g | q - 1, e >= 2, t <= L <= 40, with T_L(e) = N_F(L; e) - k^L/e, the ratio sum mu(e) T_L(e) / sum |T_L(e)| reads -0.211, -0.123, +0.069, -0.498 at L = 10, 20, 30, 40 for F = {0,1}, q = 3, and +0.812, -0.495, -0.127, -0.192 for one excluded digit at q = 10, swinging across [-1, 1] with no decay, Abs_L/k^L at 2.1 * 10^-4 and 3.9 * 10^-12 at L = 40; counts exact by the carry DP pinned against brute force and the residue DP at every reachable e <= 30000, mu from a complete certified factorisation with zero unknown cofactors. Witness: mobius.md digit strings across divisors; lab/rho-decoupling, the carry sweep and its five pinned tests.

Automata

• [Proved] Wolfram rule N and the design mrly_bang_d3_N are one subset of {0,1}^3 under (x0, x1, x2) = (l, c, r) with corner i = 4 x0 + 2 x1 + x2, so every design invariant is a rule invariant; recomputed on all 2048 rule cells. Witness: lab/automata-census.
• [Verified] The elementary rules fall into 160, 88, 22 and 14 classes under reflection, Wolfram equivalence, the cube group and the cube group with complement, by orbit walk and by Burnside alike; Wolfram equivalence and the cube group are incomparable inside the order-96 group and meet exactly in the reflection, rule 137 lying in the Wolfram class of 110 and outside its cube orbit. Witness: lab/automata-census.
• [Proved] The cube group is not a dynamical symmetry of the elementary automata: surjectivity is not a cube invariant, exactly one of the 22 classes being mixed, the 24-rule orbit of 30 with 16 surjective rules and 54 among the 8 that are not, while reversibility is constant on all 22 classes, the six reversible rules 15, 51, 85, 170, 204, 240 being the orbit of the identity, and both properties are constant on all 88 Wolfram classes. Witness: lab/automata-census.
• [Verified] The 24 rules in the cube orbit of 110 share popcount 5, degree 3, genus compound, the Walsh amplitude profile and non-surjectivity, only their single-seed diagrams separating all 24; the multiset of Walsh amplitudes by weight is a cube invariant on all 256 rules while the signed level sums are not. Witness: lab/automata-census.
• [Verified] Rules 60 and 102 draw the plane designs mrly_bang_d2_13 and mrly_bang_d2_14 cell for cell from one seed, and rule 90 draws mrly_bang_d2_13 in the sheared frame j = (t + i)/2, each unique among the fill-3 codes to level 8 by two renderers. Witness: lab/automata-census.
• [Proved] The single-seed diagram of rule 150 is an XOR substitution, row 2t being row t spread by two and row 2t + 1 that row xor its two unit shifts, and its first 2^k rows hold P(k) = 2^k F(k+2) live cells with B(k) adjacent pairs under P' = 4P - 2B and B' = 2P - 2B, so the growth exponent is log2(1 + sqrt 5); the recurrence and closed form are already A087206's and A071053's and are re-derived here. Witness: lab/automata-census, A087206, A071053.
• [Verified] One live cell under the 256 rules gives 143 distinct space-time diagrams, 89 up to reflection, class sizes {1: 118, 2: 13, 4: 4, 8: 4, 16: 4}, identical at pad T and 2T for T = 64, 128, 256; equal occurring key implies equal diagram with 0 failures over 65536 ordered pairs, the key measured on the padded line takes 152 values by a boundary artefact, and on the cropped window exactly 143. Witness: lab/automata-census.
• [Proved] In every dimension the level-1 side-3 tile of the design "not every coordinate odd" is the 3^D - 1 Moore neighbourhood and the tile of "at most one odd coordinate" fills 2^(D-1) (D + 2); as popped masks the two agree iff D <= 2, 20 against 26 cells at D = 3, and in the plane the Moore mask is mrly_bang_d2_7. Witness: lab/life-census, lab/sibling-census.
• [Verified] B3/S23 is the D = 9 design of fill 140 of 512, so Langton's lambda is 140/512 by definition, with GF(2) degree 8 on 184 monomials, Walsh level sums 140, 308, -224, -896, -168, 840, 448, -224, -196, -28 on the 0/1 form and genus compound; over the 2^18 life-like rules the fill takes 479 of 513 values with 34 gaps, the genus splits 2044 isotropic, 4 axial only and 260096 compound, exactly 8 rules are affine, and the degree obeys deg(B, S) = deg B when B = S and max(deg B, 1 + deg(B xor S)) otherwise, giving 2 * 4^d rules of degree at most d. Witness: lab/life-census.
• [Proved] The nine-cell Moore XOR B1357/S02468 is rule 150 tensor rule 150, every time slice the outer product of two rule 150 rows and its population A071053 squared, which is A246035; the named replicator B1357/S1357 is the eight-cell XOR with kernel (1/x + 1 + x)(1/y + 1 + y) - 1, A160239, and equals the outer product with the centre copy removed only at t = 2^j. Witness: lab/life-census, A246035, A160239.
• [Proved] The decoupling lemma: an automaton whose dependency set, centre included, generates a sublattice of index k is k interleaved copies of the index-1 rescaled automaton under the same rule, so the 1D parity tile at side 2r + 1 decouples iff r is even and the Cantor tower iff its level is even; over the base-2 masks at D = 1, 2, sides 3 to 9 and levels 1 to 3, the 195 distinct masks split 95 of index 1, 70 of index 2, 18 of index 4 and 12 rank-deficient. Witness: lab/sibling-census.
• [Proved] B3/S23 is not a rows-then-columns composite of two elementary rules in either order; the 65536 ordered pairs give 32260 distinct nine-input rules, none equal to Life, and exactly 10 life-like composites, all affine or thresholds at 0 or 9. Witness: lab/sibling-census.
• [Proved] Cantor-Life, B3/S23 on the eight-cell level-3 Cantor mask at offsets +-5, +-7, +-11, +-13, has no still life under 4 cells at any width, {0, 5, 7, 12} being minimal, and its nine-cell XOR has period dividing 256 on the ring of 1024 by Frobenius, exactly 256 on generic soups; every seed of width at most 14 dies (6113), stills (2003) or oscillates (76, periods 2, 3, 4, 6), and no mover is found there. Witness: lab/sibling-census.
• [Proved] Menger-Life, B3/S23 on the 20-cell Menger mask, leaks out of every plane holding a dead cell of Moore count 3, and a plane 2 x 2 block stacks as rule 90 along the normal with population 4 * 2^popcount(t); over 101 outer-totalistic rules and 200 random 5^3 seeds each, no mover appears. Witness: lab/sibling-census.
• [Proved] Under the composite 110.110 the bounding box of every finite pattern grows without bound, its upper-left corner moving (-1, -1) each generation; the population growth itself is only recomputed, not proved. Witness: lab/sibling-census.

Wallis sieve

• [Proved] The solid Wallis sieve, which drops the centre cube of every surviving cube cut into (2k+1)^3 at level k, keeps the limit volume prod_{n odd >= 3} (1 - n^(-3)) = pi^(3/2) / (8 |Gamma(7/4 - i sqrt(3)/4)|^2) = 0.948815486, by the Weierstrass product for 1/Gamma after m^3 - 1 = (m - 1)(m - w)(m - w^2) turns the k-th factor into k (k + (1 - w)/2)(k + (1 - w^2)/2) / (k + 1/2)^3 with the three shifts summing to 3/2; the plane sieve's limit area is Wallis's pi/4. Witness: mrlynum::sieve::solid_limit, evaluated by the log series to one ulp of 0.9488154857196796, checked against the truncated product to 1e-14 and against the ratio form cosh(pi sqrt(3)/2) / (3 pi) over the even product.
• [Proved] The Wallis sieve is a mixed-radix schedule word: letter k is the side-(2k+1) tile with its centre cell removed, the word is their Kronecker fold, and the word's fill is the product of the letters' fills exactly, so any odd schedule in any dimension is a sieve with area the product of its letters' survival ratios. Witness: mrlynum::sieve::ratio, the raster count against the product at levels 1 to 3 in both schedules.
• [Proved] A schedule of distinct odd letters buys area and a constant one buys a dimension: with strictly increasing odd sides sum s_k^(-d) converges, the limit area is positive and the odd word's fill exponent d + log(ratio_L) / log(side_L) walks up to d (1.972027 at plane level 4), while the constant side-s word's stands at log(s^d - 1) / log s forever (log 8 / log 3 = 1.892789 for the carpet); a schedule that merely varies, 3, 5, 3, 5, ..., loses its area and holds log 192 / log 15 = 1.941432. Witness: mrlynum::sieve::exponent.
• [Verified] The plane sieve's truncated product reads 0.785398262 at two million factors against pi/4 = 0.785398163, the identity being Wallis's. Witness: mrlynum::sieve::ratio, the test that pins its nine-digit rounding.

The arithmetic pole

• [Proved] The comb zeta of base 3 digits {0,1} has a genuine pole at every s_(0,k) = log_3 2 + 2 pi i k/log 3, k = 1..10: the residue lambda_(0,1) lies in [0.231891517689918, 0.231891517689919] + i [-0.501067414481069, -0.501067414481068] by interval arithmetic on Burnol's Proposition 5.1 with every truncation bounded by a proved tail, and Res = s_(0,k) c_k ties it to the k-th Fourier coefficient of the log-periodic profile of A(x); the profile control from exact counts and a second certified enclosure by the functional-equation route both meet the first. Witness: lab/burnol-residue, dimensions.md THE ARITHMETIC POLE.

The cocycle's joint spectral radius

• [Proved] The component cocycle of mixed Kronecker words is simultaneously triangularizable over Z and its joint spectral radius is the largest fill: in the frame (gamma, h, v, phi) = ((1,1,1,1), (1,1,2,2), (1,2,1,2), (1,2,2,4)) of components, horizontal runs, vertical runs and fill, every class matrix is nonnegative, integer and lower triangular with column sums (comp(A_c), r(c), s(c), k_c) and diagonals (0,0,0,1), (0,1,0,2), (0,0,1,2), (0,0,0,2), (1,1,1,3), (1,2,2,4), so the cross-polytope P = conv{+/- gamma, +/- h, +/- v, +/- phi} satisfies M_c P subset k_c P with exact integer residuals and its gauge is an extremal norm; hence JSR(F) = max fill and LSR(F) = min fill on all 2^15 - 1 subfamilies, the finiteness property holds with a one-letter spectrum maximizing product, and every word over {3, 6} has spectral radius exactly 2^L. Witness: connectivity.md THE JOINT SPECTRAL RADIUS OF THE COCYCLE, lab/jsr-schedules.

The repunit layer

• [Proved] The binary-weight floor on the repunits w = R_k = (3^k - 1)/2 in closed form: Phi_k = sum_{q | rad R_k} mu(q) q^(-1) sum_{t mod q} P_{q,t}^(k/ord_q 3) with P_{q,t} = prod_{r < d} (1 + e(t 3^r/q)), C-finite in k prime by prime, N_k(2) = 2^(k-1), N_k(p) = (2^k + p - 1)/p whenever 2 is a power of 3 mod p (p = 5, 7, 23), and Phi_k = 2^k - 2 whenever R_k is prime; values 2, 6, 8, 30, 24, 126, 112, 450, 460, 1958, 1344, 8190, 8064, 27360 at k = 2..15 by three methods. Witness: lab/ratio-set-saving (ratio.py repunit), coprime.md THE REPUNIT FLOOR EXACTLY.
• [Proved] The repunit excess is a lift family and a deep tail: a binary K is a multiple of R_k exactly when its column counts satisfy sum_r c_r 3^r = 0 mod R_k; below 3^(2k) the multiples are K_T = a_(T^c) + 3^k a_T with multiplier 1 + 2 a_T and R_(2k); each lift set Occ_T is occupied and stable in k, so Z(R_k) >= |union_T Occ_T|, and at prime R_k (k >= 15, first at k = 71) Z(R_k) - Phi_k >= 2^k/7 - 4 F(k+1) - 126. Witness: lab/ratio-set-saving (ratio.py repunit), coprime.md THE REPUNIT EXCESS.
• [Verified] The repunit excess Z(R_k) - Phi_k reads 0, 0, 0, 0, 0, 6, 6, 50, 70, 402, 290, 2198, 2376, 8830 at k = 2..15; the lift union equals Z(R_k) at k <= 10 and falls short by 18, 16, 108, 162, 624 at k = 11..15; minimal witnesses reach 436 digits with a column used 27 times at k = 13; the drift factorises exactly as Z(R_k)/R_k^(log 2/log 3) = 2^(log 2/log 3) (1 - 3^(-k))^(-log 2/log 3) delta_k (1 + X_k) with delta_k = Phi_k/2^k and X_k rising 0.0476 to 0.3227 over k = 7..15. Witness: lab/ratio-set-saving (ratio.py repunit).

The circle count

• [Proved] A design's corner disc count has a self-similar main term with a log-periodic multiplier: with F the base-3 digit-restricted set of a design, m its digit count, d = log_3 m and N(r) the filled cells whose centre lies in the closed Euclidean ball of radius r about the lattice corner, N is independent of the level, M(r) = m^k mu(B_(r 3^(-k))) = r^d G(log_3 r) with G positive and 1-periodic, and |N(r) - M(r)| <= C(r), the crossing count, which is O(r^(D-1)) in every D because a cell meeting the sphere lies in the shell | |y| - r | <= sqrt(D), of orthant volume 2^(-D) omega_D ((r + sqrt D)^D - max(r - sqrt D, 0)^D), so N(r) = r^d G(log_3 r) + O(r^(D-1)), an unconditional saving r^0.8927892607 on the carpet and r^0.7268330279 on the sponge; the density-times-volume main term fails outright at the grid centre, whose middle block is empty at every level and where the relative error is exactly 1. Witness: lab/circle-crop (22028 asserted rows, 6802 with a live error band, 41 mrlymath::shape::census cross-checks, the crossing bounds C_full <= 3r + 5 at D = 2 and C_full <= pi sqrt 3 (r^2 + 1) at D = 3 asserted at every radius), crop.md THE CIRCLE COUNT.
• [Verified] The crossing exponent of the corner disc count reads min 0.871371 / mean 0.898741 / max 0.969141 per triadic step over r = 27..19682 on the carpet and 1.704391 / 1.733764 / 1.757218 over r = 27..728 on the sponge, both bands containing d - 1; the defect exponent sits in [0.220478, 1.015046] (carpet) and [0.645285, 1.730726] (sponge); the powers of the base carry no resonance, ranking inside [0.1200, 0.1296] in the crossing profile at k = 3..8 on the carpet while the defect's eight ranks average 0.4788 and reach 0.8333; mu(B_1) is certified in [0.750767350, 0.751113415] on the carpet, excluding 3/4, and [0.475928750, 0.485478125] on the sponge. Witness: lab/circle-crop.
• [Proved] The crossing shell is a tree: the shell at level j is the whole grid's shell at real radius r/3^j, so it is a rooted tree of depth L with 2r+1 leaves and 2*floor(r/3^j)+1 boxes per level, and C(r) counts the leaves whose path never takes the centre seat; mean branching is 3 + (2k-2)/(2Q+1), exactly 3 at every level where floor(r/3^j) is 1 mod 3. Witness: crossing cells brute-forced from the cell definition at every level of every r <= 150 and at 18 large and boundary radii with no fault, the live leaves equal the corner ball's Cut column at every radius of every depth 1..5 for codes 7, 11 and 15, no box lacks a crossed parent at any r <= 242, and C(100) = 134 and C(242) = 296 read twice by paths sharing no code (lab/circle-crop, crates/mrlyweb/tests/shell.rs, crop.md THE CIRCLE COUNT).
• [Proved] The crossing ladder is an exact ratio of integer counts: Psi(r) = prod_k g_k(r) with g_k = u_(k+1)/(1 - p_(L-1-k)), u_k = T_k/T_(k-1) and k = L-1-j the depth from the top; g_0 = 1 identically, and g_1 = 1 exactly whenever the level-(L-1) or level-(L-2) centre box is uncrossed, so the profile carries at most L-1 informative ranks and often L-2. Witness: lab/circle-crop ladder lines, the product asserted against Psi computed directly to 1e-12 at r = 80, 242, 1000, 6560.
• [Proved] The crossing shell's transfer operator is the tripling map on the offset: with R_j = r/3^j and y_j(x) = sqrt(R_j^2 - x^2), the offset a_j(i) = frac(y_j(i)) satisfies a_(j-1)(3i) = frac(3 a_j(i)) at every level and column, because y_(j-1)(3x) = 3 y_j(x) is an identity of reals and needs no hypothesis; the 9-bit box pattern is mask(floor(u - k sigma)) for k = 0..3 clipped to [0,2], with sigma the scale-free slope. Witness: lab/circle-crop derivation pass, the derived pattern law reproducing the shell with 2 faults of 2188 boxes at level 0 and none at levels 1..5 on the shallow arc at r = 6560, the steep half following by the shell's own symmetry.
• [Proved] A 9-bit box pattern is realisable by a straight line exactly when max_(k<l) (v_k - v_l - 1)/(l - k) < min_(l<k) (v_l - v_k + 1)/(k - l) with the upper end positive; the test returns exactly thirty masks, which are character for character the crossing shell's thirty, and it forbids the thirty-first, a zero step beside a step of two, so the extra pattern seen only at the top of the tree is a curvature state and not a line state. Witness: lab/circle-crop derivation pass, stable at search radius 12 and 15 and asserted at r = 6560, 19682, 12345.
• [Verified] The Perron root of the memory-one pattern matrix differs from 3 at r = 6560, 19682, 12345, brackets [3.000861, 3.000862], [3.000948, 3.000949] and [2.997616, 2.997617], but this does not test the Markov property: the matrix row sums are exactly popcount(s), so the model returns each level mass exactly (161, 485, 1457, 4373, 13121 at r = 6560), the tree's own pooled branching is 3.037736, 3.012422, 3.004124, 3.001373, 3.000457 and never the exact 3, the certified distance from 3 falls monotonically 0.323840, 0.002623, 0.002385, 0.000086 as the parent count rises, and over 18 unused radii the sign of rho - 3 is positive 9 times and negative 9 times at sizes 0.000336 to 0.005951. Witness: lab/circle-crop operator, pair and sweep_radii lines.
• [Verified] The thirty-pattern alphabet of the crossing shell's transfer operator belongs to the truncation levels 1..L-3 and not to the tree: over levels 1..L the alphabet is 31 at r = 6560, 19682 and 12345, the extra being the root's own pattern, and 228 of the 16683 radii r = 3000..19682 leave the thirty over levels 1..L-3 and 1966 over levels 1..L-1, none reading fewer. Witness: lab/circle-crop every_level_states and scan lines, with r = 1395, 1739, 6570, 15122, 3182 pinned at four truncations.
• [Verified] Folding N(r)/r^d at 16 offsets of log_3 r mod 1 and comparing consecutive triadic windows measures the collapse instead of assuming it: the carpet's largest gaps run 0.353553, 0.222183, 0.110138, 0.042663, 0.015114 from R = 1 to R = 243 with shares of the window level 0.534078, 0.305035, 0.145250, 0.055448, 0.019546, so R = 81 and R = 243 agree to two percent and the ladder falls like 1/r; the deepest window opens at N(243)/243^d = 0.751038, inside the certified mu(B_1) bracket [0.750767350, 0.751113415], while the sponge reads 0.229755 at R = 9 against R = 27 and the grid centre does not collapse at all, its share reading 3.081886. Witness: mrlyweb crop_collapse, crates/mrlyweb/tests/crop.rs, sites/net/check.ts.
• [Proved] At D = 2, with L the least level with r < 3^L, the crossing shell's digit rate is pinned to 1/9 with an unconditional power saving and the index ind(r) = prod_(j<L) (1 - p_j(r)) (9/8)^L is bounded: the shell is a monotone lattice path, so a level-j box carries leaves(X) = w + h - 1 cells with sum_X w = sum_X h = floor(r/3^j) + r + 1 and sum_X leaves = 2r + 1, the coordinate swap makes the seat class's two marginals equal, and p_j(r) (2r+1) = 2 sum_(seat X) w(X) - #seats exactly; that turns both counts into sums of floor of one circle arc over 3^j residue classes mod 3^(j+1), where van der Corput's Satz 5 on closed subintervals [a, b] with 3^(j+1) b + c < r and a residue split of the monotone increments y(x) - y(x+1) give |p_j(r) - 1/9| <= 13.60 R^(-1/3) + 305.08 R^(-1/2) + 9.84 R^(-1) at R = r/3^j, hence sum_(j<L) |p_j - 1/9| <= 781, and with 1 - p_j >= 3^j/(2r+1) from box column 0, |log ind(r)| <= 1191 for every r >= 1; Psi is untouched, so Phi does not follow, and the staircase is a plane fact so the sponge has none of it. Witness: lab/circle-crop index lines, every identity asserted in exact integers at every level of r = 80, 242, 1000, 2186, 6560, 12345, 19682 and the bound asserted live at j = 0 for r = 212957, 531441, 2000000 where the cap clears the trivial 8/9; crop.md THE CIRCLE COUNT.
• [Proved] At D = 2, with L the least level with r < 3^L, no level-j column block [3^j i, 3^j (i+1)] of the crossing shell of radius r tracks a slope window of width eps unless 3^j < 2 eps r, so a leaf's tracked run is at most floor(log_3(2 eps r)) + 1 levels, and no level-n column block is a line's staircase once n > log_3(1 + 2 sqrt(3r)): a block's slope span t(3^j (i+1)) - t(3^j i) is at least 3^j/r because t(u) = u/sqrt(r^2 - u^2) has t' = r^2 (r^2 - u^2)^(-3/2) >= 1/r, and tracking is inherited downwards so the tracked levels are a run from the bottom and never a gap; at the Dirichlet width eps = 1/b^2 no level-j block tracks a denominator b >= sqrt(2 r/3^j), so no block at rank k from the top of the Psi ladder tracks a rational with b >= sqrt(6) 3^(k/2) and ranks 0 to 4 are held to b <= 2, 4, 7, 12, 22; the block cap is sharp, since a >= 1, ab + 1 <= b^2 and 8 * 3^(2j) b^4 < r^2 force a level-j block to track; and if the staircase floor(sqrt(r^2 - X^2)) agrees with a line's at three columns U, U + m, U + 2m in [0, r) then the second difference reads at least -1 from the line and at most 2 - m^2/r from the arc, so m^2 <= 3r, which gives (3^n - 1)^2 <= 12 r for a block but only w(X), h(X) <= 2 + 2 sqrt(3r) for a box and so excludes no box the arc enters and leaves through one side; hence every rank k < L - 1 - log_3(1 + 2 sqrt(3r)), which is L/2 - 2.14 ranks to leading order and never half of them, carries no frozen-slope resonance across a block, halving the exponent the refuted frozen route produced without bounding Psi, which stays Conjecture. Witness: lab/circle-crop track, budget, secant, blind, boxes and boxline lines, the run asserted between the two exact integer counts on all 1116 slope rows of F_30 at r = 3^L - 1 for L = 6, 7, 8, 9, run equal to the cap at 102, 91, 93, 98 of 279 slopes and to the floor at 26, 72, 66, 63, the three-point condition attained at exactly the largest level the block cap allows at each radius, and at r = 19682 level 6 exactly 3 of the 53 boxes with their whole content equal to a line's staircase, (16, 20), (19, 19) and (20, 16), the witness that the box statement fails; crop.md THE CIRCLE COUNT.

The codebook

• [Proved] Under a minimum-description-length score with a catalog of A = 667 atoms and at most P = 59049 positions, the name charge log2 A + log2 P kills every atom below 31 cells outright, 131 of the 667, while every surviving atom still pays its share of log2 A; the arithmetic of the score, not a measurement. Witness: lab/codebook.
• [Verified] The rearranged SVD of a level-5 carpet render returns its level-1 code 495 exactly at four block splits (sigma_2/sigma_1 <= 3.3e-15) and peels the five-letter magic word carpet(3), void(3), net(3), htree(3), vtree(3) into 495, 341, 186, 455, 365; under bit flips at 51 rates and 40 seeds each the code survives 40/40 to p = 0.30, 12/40 at 0.31, 0/40 from 0.32, against a block-mean baseline whose closed-form failure f/(1 + 2f) = 0.277638, f = (8/9)^4, is derived from the fill law and not from the sweep. Witness: lab/codebook.
• [Verified] The three-class codebook (Kronecker tiles, magic designs, life frames) under the MDL score loses to raw deflate on the tree's own render by 17212 bits, on text by 31375 and on a halftone by 31217 at Kronecker level 2, edges it by 29 bits on random bytes where deflate itself expands the stream by 47, and crosses deflate on the render only at catalog level 3; the uniform-bit control takes zero placements and saves -18 bits at every catalog depth 2 to 5, and every corpus reconstructs bit for bit from placements plus residual. Witness: lab/codebook.

The moments of the digit transform

• [Proved] The even moments of the digit transform are additive energies counted by a carry DP: sum_{a mod q^L} |hat F_L(a/q^L)|^(2r) = q^L E_r(L), E_r the r-fold additive energy modulo q^L of the length-L strings, C-finite in L of order at most r(r+1)/2 with growth constant Lambda(2r) = q rho, rho the certified Perron root of the carry-pair transfer matrix; Lambda(4) = 18 at {0,1} base 3 (x - 6), 2(23 + sqrt 353) at {0,1,2} base 4, (275 + 5 sqrt 2369)/2 at {0,1,2,3} base 5, every value strictly inside [max(k^4, q k^2), q k^3]; brute force at L <= 7, direct grid evaluation at L <= 6, the bounds and the recurrence asserted to L = 60. Witness: lab/rho-decoupling (the riesz module, 19 tests), mobius.md THE METER AND ITS YARDSTICK.
• [Proved] The multiplicative energy E_x(L) = #{n_1 n_2 = n_3 n_4} of a digit-restricted column has exponent 2 alpha for every base and digit set: 2K^2 - K <= E_x(L) <= K^2 max_m r(m) with r(m) <= d(m), so E_x(L) = k^(2L) x^(o(1)); the census reads 1, 15, 111, 655, 3179, 14211, ... to 58760487 at L = 1..12 for {0,1} base 3 with theta_x = 1.475642, 1.410978, 1.356938 at L = 4, 8, 12 falling toward 1.261860; the shift family (q^i u, q^j v, q^(i') u, q^(j') v), i + j = i' + j', i != i', counted in closed form in k and L when 0 is a digit, is a floor on the excess over the two diagonals, 0.4418 of it at {0,1} base 3, L = 12 and 0.19 to 0.0003 at the other families. Witness: lab/rho-decoupling (the menergy module, 27 tests), mobius.md THE METER AND ITS YARDSTICK.

Weighted designs

• [Proved] For a weighted design (a probability vector on |F| >= 2 cells) the Dirichlet root of sum_f w_f^s = 1 is identically 1, so it is the arithmetic class of the log w_f, never the root, that carries the mass-stopping count N(t) = #{words of mass >= t}: N is log-periodic exactly when the group generated by the log w_f is cyclic (for rational weights, when the prime-exponent matrix has rank 1) and smooth otherwise, by Lalley's renewal dichotomy on f = -log w; and every length observable keeps its log q ripple at every weight, mu(B(r)) = w_0 mu(B(qr)) at a corner fixed point for r < min_{f != 0} |f|/q. Witness: lab/weighted-designs.
• [Verified] The multifractal pressure of a weighted design with equal contraction 1/q under the open set condition is tau(s) = log_q sum_f w_f^s, so f(alpha) = inf_s (alpha s + tau(s)) is explicit: the level-L box moments carry it exactly as sum_i mu_i^s = (sum_f w_f^s)^L, the coarse-grained band sits under the transform at every level (f_L <= f from N_i mu_i^s <= q^(L tau(s))), exact at both endpoints and deficient by 0.176458, 0.147536, 0.127619 at the band's middle alpha = 1.077324384 at levels 6, 8, 10 on the three-cell weighted gasket (base 3, cells (0,0) (2,0) (0,2), weights 3/8, 3/8, 1/4), the deficit matching Stirling's series, and the JSR bracket validates on the hat mask at alpha = 1 and on D4 at alpha in [0.4929285, 0.5500157] against the closed form 2 - log_2(1 + sqrt 3) = 0.5500157. Witness: lab/weighted-designs, Cawley and Mauldin 1992.

Mertens stack

• [Verified] Weighting the Farey stack by mu(n) renders a Mertens-type sum at every node, Sum_{k <= N/b} mu(kb) at a/b and M(N) at b = 1, and the log-space power spectrum of M(x)/sqrt(x) shows the first eight nontrivial zeta zeros, detected at 13.94, 20.90, 24.97, 30.19, 32.52, 37.74, 40.64, 42.97 against 14.1347, 21.0220, 25.0109, 30.4249, 32.9351, 37.5862, 40.9187, 43.3271, errors 0.04 to 0.42 inside one bin of width 0.5806; the zeros are known to far beyond any precision a moire can reach, so this is a rendering and not a measurement. Witness: lab/mertens-meter.

Complex dimensions

• [Proved] The Sierpinski carpet is not Minkowski measurable: its complement in the open square is the disjoint union of 8^(m-1) open squares of side 3^(-m) whose boundaries lie in the carpet, the tube is the exact hole sum, and eps^(d-2) V(eps) -> G(t) with G = t^(d-2)(1 + 4t/5 - 4t^2/7) on [1/3, 1/2) and t^(d-2)(9/8 + 3t/10 - t^2/14) on [1/2, 1), C^1 at the seam, 379/280 at the ends, maximum 1.35561708227 at t = 0.429638, minimum 1.3506702097 at t = 0.692137, swing 0.3662%; a corollary of Kombrink, Pearse and Winter 2016 Theorem 1.1(ii), whose hypotheses are verified for the carpet with the open square, the profile and the elementary proof being the addition. Witness: lab/complex-dimensions carpet_tube.py, dimensions.md measurability with its hypotheses.
• [Proved] Every one-base design at base q >= 3 in D >= 2 removing at least one digit vector, all removed vectors interior and pairwise differing by at least 2 in a coordinate, is not Minkowski measurable: q^(D-1) < k < q^D so d = log_q k is never an integer, G(t) = t^(d-D) sum_j k^(j-1) q^(-jD) h(t q^j) > 0, and t^(D-d) G is a polynomial on [1/q, 1/2); includes the parity carpets at every odd base and dimension. Witness: dimensions.md measurability with its hypotheses (proof), lab/complex-dimensions at q = 3, D = 2 only.

Coprimality at dimension one

• [Proved] At k = 2t+1 the lift T = [t, 2t-1] has multiplier m_T = 3^(2t) - 3^t + 1 = Phi_6(3^t) dividing the binary 3^(3t) + 1, so K_T carries at least 2^t submasks divisible by m_T against an equidistribution model below 1; no uniform bound of the shape C 2^k / m_T^c survives c > log 2 / (2 log 3) = 0.3154649 while Sum_T m_T^(-c) converges only for c > log 2 / log 3 = 0.6309297, so every exponent that would close the lift-union half is refuted for that shape. Witness: lab/ratio-set-saving, ratio.py check 2.5 s and ratio.py lifts --kmax 19 21 s.
• [Proved] Antipodal lift family: for odd p, 0 <= s <= t, k = (p-1) t + s, the set T = Union_(i odd) [ti, ti + t - 1] has m_T = (3^(pt) + 1)/(3^t + 1) and exactly 2^(((p-1)/2)(t - s) + s) submasks of K_T divisible by m_T, by antipodal pairs, blocks and balanced-ternary uniqueness. Witness: lab/ratio-set-saving, ratio.py agg asserts all 74 triples to k = 19, check at k <= 8.
• [Proved] A binary K with support inside [0, bk - 1] is a multiple of R_k exactly when its column counts satisfy Sum_r c_r 3^r = j R_k, and for b <= 3 that forces the column vector constant, so the binary multiples of R_k below 3^(3k) are exactly 2 * 3^k + 1 lifts, 3^k with one position per column and multiplier 1 + 2 a_(E_1) + 2 (3^k + 1) a_(E_2), 3^k with two, and R_(3k), which yields only submask directions; at b = 4 the column vector branches, 24 non-constant vectors at k = 3. Witness: lab/ratio-set-saving, ratio.py tail and ratio.py check.

Primes on a design

• [Proved] Primes on a design at k = q^D - 1 are primes with one restricted digit at base q^D: the Morton code x -> Sum_j (Sum_c q^(c-1) x_(c,j)) q^(Dj) maps S_n bijectively onto the base-q^D integers whose digits lie in the image of F, the gasket base 4 missing 3 and the carpet base 9 missing 4; the gcd-prime reading is a positive-density count when B(F) > 0 and empty otherwise, base 32 on {0,4,...,28}^2 having k = 64 > 32, (E), and every gcd divisible by 4; the x_1-prime reading is a sum-of-digits large deviation. Witness: coprime.md PRIMES ON A DESIGN.
• [Proved] Lemma A' the window rate: for (E), gcd(d,q) = 1 and nonzero t in (Z/d)^D, Prod_(l<n) f_l(t) <= c(q,k)^floor(n/m_d) with m_d = max(1, floor(log_q(d/2)) + 1), and ord_d(q) >= m_d so it is never weaker than Lemma A; with a base part e and t nonzero mod the coprime part m, and |eta|_inf < q^(-2n/3)/(4qD(q-1)), the rate is c'(q,k)^floor(2n/(3 m_d)), which is Maynard Lemma 8.2 in every dimension with an explicit constant and no consecutive-digit hypothesis; the hypothesis on t is sharp, the gasket at d = 6 and t = (3,0) sitting at 1/3 at every n. Witness: lab/digit-transform-norms lemma, worst per-digit rate 0.830915 at d = 257 over d <= 301 against Lemma A's 0.986514.
• [Proved] The 2D Type I saves a power when alpha_1* < D/2, the dyadic block d ~ Q_1 costing k^n (Q_1^(2 alpha_1* - D) + Q_1^D q^(n(alpha_1* - D))) and the small moduli going to Lemma A': the carpet certified at alpha_1* < 0.8124 gives Sum_(d <= Q, gcd(d,3) = 1) |#{x in S_n : d | x} - k^n/d^2| <<_A k^n n^(-A) at Q = 3^(0.5938 n) n^(-C), and the gasket certified at alpha_1^- >= 1.0126, alpha_1^+ <= 1.1022 closes the route, min_x Sigma_2 > 4.059204 against 4 and min_x Sigma_3 > 8.213932 against 8 in interval arithmetic with directed rounding, Sigma_2(0) = (8 + 2 sqrt(5))/3 exactly. Witness: lab/digit-transform-norms certify.
• [Verified] The carpet misses the one-dimensional criterion at every order: base 9 missing 4 has g(1) = 0.3437 below 27/77 but g(3/2) = 0.1531, g(235/154) = 0.1457, g(1.6) = 0.1262, g(1.7) = 0.1031, g(1.8) = 0.0835 against 0.1473, 0.1397, 0.1179, 0.0884, 0.0589, a gap of 0.0058 on the printed pair at s = 3/2 and 0.005749 in full, and g(3/2) moves 0.154389, 0.153068, 0.152921 over four, five and six digit-vectors, so windows do not close it. Witness: lab/digit-transform-norms moments.
• [Verified] The gasket is out of reach at both numbers: base 4 missing 3 has g(1) = 0.4820 against 27/77 and g(235/154) = 0.3170 against 59/433, so no Type II range opens at any order computed. Witness: lab/digit-transform-norms moments.
• [Verified] The componentwise route is closed at source: Chow, Varju and Yu Remark 6.1 puts the Fourier l^1 dimension below 1/2 for (b,a) in {(3,0),(3,1),(3,2),(4,1),(4,2)} by interval arithmetic at L = 2, so the base-3 design's coordinate marginals fall on the wrong side, while Proposition 2.4 puts base 4 missing 3, the base-2 gasket's Morton code, above 1/2. Witness: arXiv:2402.18395v2 pp.25-26.
• [Verified] The missing-digit criterion is unreachable for the carpet at every order: dividing by 2 - s the criterion is the single inequality g(s)/(2 - s) < (1/5)*(1 + c/2) on the transform's moment exponents, and for base 9 missing 4 the shift sandwich at a power certifies g(3/2) > 0.149397 and g(235/154) > 0.142274 against the required 0.147320 and 0.139667, with a monotone chain of orders anchored at the exact Sigma_N^(2)(x) = (9/8)^N covering [3/2, 2) in 21 closed cells sharing endpoints and [1, 2) in 87; the pointwise deficit is at least 0.001268 over [3/2, 2) and the decisive cell re-derived independently at N = 5 clears by 0.000840. Witness: lab/digit-transform-norms criterion, with an independent recomputation by a digit-tree fold reproducing every printed digit.
• [Proved] The two-missing-digit transform is (q-2)^2 abs(hat F)^2 = K^2 + 2 + 2 cos(2 pi D t) - 4 K cos(pi S t) cos(pi D t) with K = sin(q pi t)/sin(pi t), D = a - c, S = a + c - (q-1), so a pair enters only through abs(D) and abs(S); that implication does not run backwards, {0,2} and {0,8} at q = 10 reading (2,7) and (8,1) with equal transforms, and the collapse is generated instead by the reflection d -> q-1-d, which flips both signs, together with the integer translation of F available exactly when 0 or q-1 is excluded and identifying {0,c} with {0,q-c}, so the edge family is the one-missing-digit sets of a (q-1)-digit interval read at base q and the number of distinct transforms is (C(q-2,2) + floor((q-2)/2))/2 + floor(q/2). Witness: coprime.md PRIMES ON A DESIGN.
• [Verified] That pair count reads 7, 16, 21, 31 of the 15, 36, 45, 66 excluded pairs at q = 6, 9, 10, 12, is reproduced by grouping every one of the C(q,2) pairs by its sampled transform at every base 4 <= q <= 41, and sums to 2373 distinct sets over 4 <= q <= 31. Witness: lab/digit-transform-norms pairs.
• [Verified] The least base carrying a certified two-missing-digit set with alpha_1 < 1/4 is q = 32 at the interval class {0,1}, alpha_1 in [0.2499087, 0.2499779] at four window digits, the same class at q = 31 reading [0.2518967, 0.2519717]; over 4 <= q <= 31 the machine certifies alpha_1 > 1/4 at 2363 of the 2373 distinct sets, closest q = 26 missing {2,23} at > 0.2502919, and the ten it cannot bracket from below all have S = 0 or D = q/2 with q/2 odd, a shared shape and not a cause since the clearing headline q = 32 missing {0,1} has a transform vanishing at all 29 points t = j/30, with certified upper bounds 0.2538899 to 0.2826357, above 1/4. Witness: lab/digit-transform-norms pairs and pairfail.
• [Verified] Against the bar 1/3 a two-missing-digit set first clears at q = 13, the interval class at alpha_1 < 0.3318819 on three window digits with q = 12 above at all 31 of its sets to five, and the whole pair family clears from q = 21 on through q = 26, worst q = 23 missing {4,5} at < 0.3333284, every base 4 <= q <= 20 carrying a certified witness above 1/3, q = 20 by {3,11} at [0.3356579, 0.3356674]. Witness: lab/digit-transform-norms pairs pairclear pairsome.
• [Proved] The digit-uniform bound holds at any excluded-digit count: abs(hat F(t)) <= (abs(sin(q pi t)/sin(pi t)) + m)/(q - m), the level product expands with weight m^(N - card E) and telescopes to the same Dirichlet kernels, so a_N = m a_(N-1) + m Sum_(l<N) lambda_l a_(N-1-l) + lambda_N and the growth root solves (z - m)(z - 1)^2 = m(c_1 (log q) z + gamma'(z - 1) + c_1 (z-1)^2/(q z - 1)) on the exact Lebesgue input lambda_l <= c_1 l log q + gamma' + c_1 q^(-l), gamma' = (2/pi)(gamma + log(8/pi)), giving alpha_1 < 1/4 for every q >= 649 at m = 2 with the chain failing at 648, and 125 at m = 1 and 1873 at m = 3, with 32, 105, 230 against 1/3, certified at 120 bits; the coarser c_0 = 0.97 form of the same chain needs q^l >= 86 and gives 126 at m = 1. Witness: lab/digit-uniform-bound pairs.
• [Proved] The threshold 1/4 is the Mertens bar: in the GRH chain steps 1, 2, 4 and 5 never name the digit set and only step 3 substitutes a digit-free kernel bound, so feeding the certified l^1 exponent there gives abs(M_F(x)) <<_(q,eps) A_F(x) x^(alpha_1 - 1/4 + eps), that is A_F(x)^(1 - delta + eps) with delta = (1/4 - alpha_1)/alpha_q > 0, and 1 - b(a) in place of 1/4 under a zero-free half plane; steps 2 and 3 alone force alpha_1 <= 1 - alpha_q + c_q and gap_q(1) > 0 is exactly 1 - alpha_q + c_q < 1/4, so the old certificate implies the new condition and the wall can only fall. Witness: coprime.md PRIMES ON A DESIGN.
• [Verified] That wall falls from 3690 to 34, on the interval certificates behind the one-missing-digit clearance alpha_1 < 1/4 at q = 34, at every 35 <= q <= 125 and by the uniform chain above. Witness: coprime.md PRIMES ON A DESIGN.

Digit designs and the Euler product

• [Proved] The indicator of S_F is multiplicative exactly at the full digit set. 1 in F is forced by f(1) = 1; if a digit c >= 2 is missing take the least, and R_c R_(c+1) has no carry because its q^m coefficient is min(m+1, c, 2c-m) <= q-1, so its digit set is exactly {1..c} while gcd(R_c, R_(c+1)) = R_1 = 1; if only 0 is missing then odd q gives the coprime pair (2, (q^2+1)/2) with product q^2 + 1 = 101, and even q gives (q^2-1, q^2+1), coprime and odd, whose product q^4 - 1 has every digit q-1 while q^2+1 does not lie in the set. Over all 8177 sets with 2 <= q <= 12 the constructed witness is asserted at each of the 4083 sets that pass f(1) = 1 and are not full, and an independent search finds a minimal witness for every one, hardest q = 12, F = {1}, pair (5, 377). No design outside the full set carries an Euler product over primes; 0 excluded and a single digit both fail. Witness: lab/mrly-euler verb wall.
• [Proved] For every F strictly inside {0..q-1} the design zeta and the design Mobius series obey a disjunction and not a universal: if 1 is outside F the constant coefficient of zeta_F M_F is 0; if a prime p of S_F has p^2 outside S_F the coefficient at p^2 is -1, since (p,p) is the only admissible factorisation; and otherwise zeta_F M_F = 1 forces the least element g > 1 of S_F to be prime with every power g^j in S_F, a necessary condition on an escapee and not a contradiction. At the full digit set the two are inverse, zeta_F = zeta and M_F = 1/zeta. Over 257 sets the least n > 1 with a nonzero coefficient is at most 50, first at n = 4 for base 3 {0,1} and n = 9 for base 10 missing 9, while the eight full sets have none below 4000. Witness: lab/mrly-euler verb pair.
• [Proved] The position product. With G_L(t) = prod_(i<L) sum_(d in F) e(d q^i t) = k^L hat F_L(t), uniqueness of the base-q expansion gives int_0^1 G_L(t) e(-nt) dt = 1_(D_L)(n) for every integer n, hence sum_(n in D_L, n >= 1) a(n) n^(-s) = int_0^1 G_L(t) A(s,t) dt for every absolutely convergent Dirichlet series, with A(s,t) = sum_(n >= 1) a(n) e(-nt) n^(-s); a = 1 is the periodic zeta of DLMF 25.13.1 and a = mu the Lerch-Mobius series, so zeta_F and M_F are pairings of one set-only product against one arithmetic-only kernel. The set enters through the digit positions and never through the primes. Checked to 1.95e-16 and 2.04e-16 at base 10 missing 9, L = 3 and L = 4, and 2.9e-16 at base 3 {0,1}, L = 3, 4, 5. Witness: lab/mrly-euler verb position.
• [Proved] The tree's pair route is Holder on the position identity. When 0 is in F, at x = q^L the identity is finite on both sides, M_F(q^L) = int_0^1 G_L(t) S_L(t) dt with S_L(t) = sum_(n < q^L) mu(n) e(-nt), so abs(M_F(q^L)) <= (int_0^1 abs(G_L)) max_t abs(S_L) is at most k^L q^(L(alpha_1 - 1)) x^b = x^(alpha + alpha_1 - 1 + b); when 0 is outside F the same upper bound holds after summing the levels, a geometric sum of ratio q^(alpha + alpha_1 - 1 + b) > 1 by the floor alpha + alpha_1 >= 1 of mobius.md. It sits under the trivial x^alpha exactly when alpha_1 < 1 - b, which is the bar of coprime.md and mobius.md derived rather than posited, with b = 3/4 + eps under GRH from Baker and Harman 1991. Witness: lab/mrly-euler verb position.
• [Proved] The fibres of the Lerch-Mobius series are inverse Dirichlet L-functions. Splitting n by g = gcd(n,Q) and expanding on the characters of (Z/(Q/g))^* gives M(s, a/Q) = sum_(g divides Q) mu(g) g^(-s) phi(Q/g)^(-1) sum_(chi mod Q/g) tau_a(chi) L(s,chi)^(-1) prod_(p divides Q not Q/g) (1 - chi(p) p^(-s))^(-1), so M(s, a/Q) continues to C with singularities in Re s > 0 only at zeros of L(s,chi) of modulus dividing Q, and M(s,0) = 1/zeta(s). Since G_L(a/q^j) = k^(L-j) G_j(a/q^j) are the largest values the position product takes, the design's major arcs are the q-power rationals, and on that family holomorphy in Re s > 1/2 is exactly GRH for q-power modulus. Coefficient identity checked to 2.6e-12 at eleven pairs (Q,a) including Q = 3, 9, 27, 100, the Euler-factor step to 7.4e-16. Witness: lab/mrly-euler verb fibre.
• [Proved] The reflection moves the kernel and not the design. Solving Hurwitz's formula DLMF 25.13.3 at x = t and x = 1-t gives Z(s,t) = ((2 pi)^s Gamma(1-s)/(2 pi i))(e^(pi i s/2) zeta(1-s,t) - e^(-pi i s/2) zeta(1-s,1-t)) for s not a positive integer, the derivation dividing by 2i sin(pi s); this is DLMF 25.13.2 recovered, the gain being the range Re s > 0 in place of Re s > 1. The position identity turns it into a dual integral of the same G_L against Hurwitz zetas at 1-s, never a relation between zeta_F(s) and zeta_F(1-s); the design's own symmetry is the q-adic scaling G_L(t) = g(t) G_(L-1)(qt), whose transfer eigenvalue k q^(-s) is what makes the vertical pole lattice. Formula checked to 2.1e-30 at s = 3.3, 2.7 + 1.9i and 0.6 + 4.1i. Witness: lab/mrly-euler verb dual.
• [Proved] The design's multiplicative shadow is a Lyndon Euler product with no RH content. On the free monoid over F with norm N(w) = q^(abs(w)), sum_w N(w)^(-s) = 1/(1 - k q^(-s)) = prod_(L>=1) (1 - q^(-Ls))^(-c_k(L)) with c_k(L) the Lyndon count, by Chen-Fox-Lyndon: every word factors uniquely as a non-increasing product of Lyndon words, so the free monoid on F is equinumerous by norm with the free abelian monoid on Lyndon words and is not equal to it. The primes are the Lyndon words, the zeta is zero-free, its Mobius is supported on the empty word and the letters so its Mertens is 1 - k beyond norm 1, and its poles are exactly s = alpha + 2 pi i m / log q, the design pole lattice. All RH content of zeta_F therefore sits in the cofactor zeta_F(s)(1 - k q^(-s)). Expansion verified through u^16 at k = 2, 3, 4, 9, 10, c_2(L) being A001037. Witness: lab/mrly-euler verb word, A001037.
• [Proved] The Beurling system of a design with non-unit digit gcd is finitely generated, on two branches. If gcd(F) = a > 1 every element of S_F is a multiple of a; when a is prime the primes of the design are {a}, N_F is the powers of a and M_B(x) = 0 for x >= a, and when a is composite S_F holds no prime at all, N_F = {1} and M_B is identically 1, witness q = 10, F = {0,4,8}. Either way the eight scaled census families of mobius.md are exactly the columns the Beurling route cannot see, while the scaling transfer reads them exactly. Witness: lab/mrly-euler verb beurling.
• [Verified] The Beurling census on the primes of a design, to x = 10^6. Base 3 {0,2} has the single prime 2 and M_B identically zero past 2; base 3 {0,1} has 525 primes, N_F(920483) = 2198, running max abs(M_B) = 98 and exponent 0.3339 against alpha/2 = 0.3155; base 10 missing 9 has 35139 primes, N_F(10^6) = 488864 against x^alpha = 531441, M_B(10^6) = 1860, running max 1866, and exponent log(running max)/log x reading 0.4203, 0.4882, 0.5452 at 10^4, 10^5, 10^6 against alpha/2 = 0.4771, where full base 10 as control reads 0.4084, 0.4241, 0.4276 at the same points against its own alpha/2 = 0.5. What the census reads is the level and not a trend: +0.068 over alpha/2 for the design against -0.072 for the control, a running maximum climbing in both. There is cancellation, 0.545 against the trivial alpha = 0.954, and it is above alpha/2, so the census supports cancellation and does not support the square-root conjecture on N_F; N_F is not S_F. Full base 10 reproduces -23, -48, 212 at 10^4, 10^5, 10^6, A084237. Witness: lab/mrly-euler verb beurling, A084237.
• [Proved] The identity that replaces zeta M = 1 on a design. For every (q,F) with 1 in F the indicator 1_(S_F) has a Dirichlet inverse nu_F, given by nu_F(1) = 1 and nu_F(n) = -sum_(d divides n, d > 1, d in S_F) nu_F(n/d), so zeta_F(s) N_F(s) = 1 with N_F(s) = sum nu_F(n) n^(-s); the support of nu_F lies inside the multiplicative semigroup generated by S_F and strictly inside it, since 9, 27 and 36 lie in the semigroup with nu_F = 0 while 16, 48 and 52 lie in the semigroup and outside S_F, so the semigroup is a third set beside S_F and the Beurling integers on the primes of the design and the support is a fourth, and nu_F is mu exactly at the full digit set, where the classical identity is the special case. If rho is a zero of zeta_F with Re rho > alpha then sigma_c(N_F) >= Re rho, by the identity theorem on the connected pole-free half plane Re s > max(sigma_c(N_F), alpha), so sum_(n <= x) nu_F(n) is not O(x^(Re rho - eps)) for any eps > 0; the converse bound sigma_c(N_F) <= sup Re rho is not claimed. Checked against mu term for term on the full digit set to n = 131072 at q = 2 and n = 177147 at q = 3, and the partial sums of N_F(sigma) meet 1/zeta_F(sigma) to 1.60e-3 at sigma = Re rho + 0.08 = 0.8008 and 1.96e-4 at sigma = Re rho + 0.20 = 0.9208 at base 3 {0,1}, and to 1.72e-2 at sigma = 1.0816 and 2.39e-3 at sigma = 1.2016 at base 10 missing 9, both offsets sitting above Re rho. Witness: lab/mrly-pairing verb inverse, lab/design-zeta.
• [Proved] The design's own Mobius has anti-cancellation, and that is what makes the decoupling a blessing. Winding boxes on zeta_F by the argument principle certify one zero each and pin Re rho to the box edges: winding 1 on Re in [0.72074, 0.72084], Im in [28.60563, 28.60573] at base 3 F = {0,1} with contour minimum abs(zeta_F) = 8.298e-4 against the engine bound 6.284e-30, and winding 1 on Re in [1.00150, 1.00168], Im in [2.73915, 2.73925] at base 10 missing 9 with contour minimum 6.865e-4 against 2.798e-23, while the control rectangle Re in [0.99900, 1.00050], Im in [2.73810, 2.74030] there returns winding 0. Both boxes lie strictly right of alpha = 0.6309297536 and 0.9542425094, so sum_(n <= x) nu_F(n) is not O(x^(0.72074 - eps)) and not O(x^(1.00150 - eps)) respectively: the limsup of the design's own Mertens function exceeds the design's own mass A_F(x), and at base 10 missing 9 exceeds x itself, the box lying right of Re s = 1. The square-root conjecture in the alpha/2 shape is therefore false for nu_F and can only be carried by mu restricted to S_F; the sibling's decoupling theorem is what protects it. Pointwise the census is far below both limsups, max/A_F = 0.0738 at base 3 L = 16 and max/x = 0.0847 at base 10 L = 7: the running maximum of sum nu_F(n) grows by 9.4474, 11.5000, 10.2220, 10.0354 per level at base 10 missing 9, L = 4..7, against q^(Re rho) = 10.036661 and the trivial k = 9, only the last of the four landing on the predicted rate, with max/A_F(q^L) rising 0.1043, 0.1094, 0.1398, 0.1588, 0.1771; at base 3 {0,1} the geometric mean of the four steps L = 12..16 is 2.059 against 2.207512 and 2 while the arithmetic mean of the five printed level ratios is 1.9972, below the trivial 2, a census too short to separate them. Witness: lab/mrly-pairing verbs box and inverse, lab/design-zeta.
• [Verified] The pair zeta_F M_F = 1 + D_F gains nothing: D_F has abscissa exactly alpha. Absolute convergence of zeta_F^2 puts sigma_a(D_F) <= alpha and that half is proved; for the other half, if sigma_c(D_F) were below alpha then M_F(sigma) = (1 + D_F(sigma))/zeta_F(sigma) would tend to 0 as sigma -> alpha+, since zeta_F has nonnegative coefficients and is singular at its abscissa by Landau, so zeta_F(sigma) -> +infinity, and there is no circularity in the argument because M_F is dominated termwise by zeta_F and so converges absolutely at every sigma > alpha with no hypothesis on theta(F). That half rests on a measurement, unconditional in shape since sigma_c(D_F) < alpha would force P(x) = o(x^alpha): P(x) = sum_(n <= x) c_F(n) divided by x^alpha is bounded away from 0 and from infinity, reading 0.493767, 0.699235, 0.758519, 0.587055 at four sampling phases at base 3 {0,1}, the four phases being needed because P(x)/x^alpha is log-periodic and sampling only at x = q^L aliases every Fourier mode onto one number. The M_F(sigma) -> 0 limit test is not a witness here: the tail the generator prints beside it is q^(-L alpha/2), which assumes the square-root conjecture, and against the unconditional tail (k-1) q^(-L eps)/(1 - q^(-eps)) from A_F(q^l) = k^l no printed M_F value at base 10 missing 9 is distinguishable from 0. Since M_F = (1 + D_F) N_F and sigma_c(N_F) > alpha, the glue is not neutral but lossy. Witness: lab/mrly-pairing verb glue.
• [Proved] The position pairing is exact on the grid and its l^1 mass sits at the top level, which kills the per-denominator split. For 0 in F, M_F(q^L) = q^(-L) sum_(a mod q^L) G_L(a/q^L) S_L(a/q^L) exactly, both factors being trigonometric polynomials of degree below q^L; writing a = q^v a' with q not dividing a' and j = L - v gives G_L(a/q^L) = k^(L-j) G_j(a'/q^j) and the exact level decomposition C_L = sum_(j=0)^L k^(L-j) c_j of the l^1 mass, with C_j = k C_(j-1) + c_j. The l^1 floor C_j >= q C_(j-1) forces the top-level share c_L/C_L >= 1 - k/q = m/q at every base and digit set, measured 0.485846, 0.602606, 0.687994, 0.510055 against floors 0.333333, 0.500000, 0.600000, 0.100000, with levels j >= L/2 carrying 0.995116, 0.996061, 0.997043, 0.942350. Since the Baker-Harman Proposition beats the uniform x^(3/4) only below j = L/2, weighting the Mobius input per denominator saves exactly log_q(C_L/c_L)/L, a constant factor capped by q/m: the numerator is 0.657068 at base 3 {0,1}, identical at every L = 6..14. The split exponents are 0.988106, 0.912502, 0.905006, 1.012881 uniform and 0.941173, 0.879287, 0.879188, 0.964150 per denominator against alpha = 0.630930, 0.500000, 0.430677, 0.954243, while the Cauchy-Schwarz split is (alpha+1)/2 exactly since int abs(G_L)^2 = k^L; the base 2 and base 3 full-set controls return 0.500000, the classical RH exponent. Witness: lab/mrly-pairing verb split.
• [Proved] The principal fibre of the grid pairing has exponent alpha - 1/2 under RH, below the conjectured alpha/2, and that is an asymptotic statement only. The a = 0 term of the grid pairing is q^(-L) k^L M(q^L), of exponent alpha - 1/2 under RH, and alpha - 1/2 < alpha/2 for every alpha < 1, so in the limit the classical Mertens function cannot carry the conjectured size of the design meter. At finite depth it carries a great deal: the a = 0 term reads -0.31857 of 11, 0.11133 of 6, -0.05924 of 9 and 112.66549 of 276 at base 3 {0,1} L = 14, base 4 {0,1} L = 11, base 5 {0,1} L = 9 and base 10 missing 9 L = 6, shares -0.028961, 0.018555, -0.006583, 0.408208, and exactly all of the meter on the two full-set controls. So at base 10 missing 9 the principal fibre carries 40.8 percent of the meter at the only measured L, which refutes any claim that the square-root conjecture lives entirely off the principal fibre at finite depth: the exponent gap there is 0.454243 against 0.477121, and a factor of 10 between them needs x = 10^44. Witness: lab/mrly-pairing verb split.
• [Proved] The one-step constant of the digit transform never exceeds the triangle-split bound, and is strictly below it at every family measured beyond L = 1. With H(t) = sum_(r mod q) abs(g_F((t+r)/q)) and B_q(F) = sup_t H(t), the identity C_L = sum_(a mod q^(L-1)) abs(G_(L-1)(a/q^(L-1))) H(a/q^L) gives C_L <= B_q(F) C_(L-1), so C_L/C_(L-1) <= B_q(F) at every L and every family with no computation at all; the inequality is not strict in general and equality is attained, C_1/C_0 = 4 = B_q(F) exactly at base 3 {0,1}, so strictness needs L >= 2. Verified there: C_L/C_(L-1) reads 3.889888518, 5.032783116, 6.410132461, 18.369402635 at base 3 {0,1}, base 4 {0,1}, base 5 {0,1} (all L = 9) and base 10 missing 9 (L = 6), against B_q(F) = 4.000000000, 5.226251860, 6.472135955, 19.888543820, and the ratio agrees between the two consecutive L the generator prints to 8.5, 7.4, 10, 5.0 digits by family, so the stability is family by family and two values of L are all that is measured. Witness: lab/mrly-pairing verb split.
• [Proved] The design Mobius of the two-digit design is base-free. Let S* be the nonzero 0/1 polynomials of Z[x], M* the monoid they generate, nu* the Dirichlet inverse of 1_(S*). For F = {0,1} at every base q >= 2, nu_F(n) = sum over P in M* with P(q) = n of nu*(P). Evaluation is a bijection S* -> S_F, a monoid homomorphism, and of finite fibres, since an element of M* has nonnegative coefficients so P(q) = n caps every coefficient by n and deg P by log_q n; the pushforward g therefore exists, g(1) = 1 because 1 is the only element of M* of value 1, and grouping the pairs (D, Q) in S* x M* with D(q) Q(q) = n by P = DQ turns 1_(S*) * nu* = delta into 1_(S_F) * g = delta, where the Dirichlet inverse is unique. Hence sum_(n <= x) nu_F(n) = sum over P in M* with P(q) <= x of nu*(P) at every x: the base enters only as the order in which one base-free function is summed, and at q = 2 the classical mu is that pushforward. Checked term for term with 0 mismatches to n <= 2^15, 3^10, 4^8 and 5^7, where 108978 elements of M* collapse onto 32768 integers at base 2. Witness: lab/carry-free-mobius verb lemma.
• [Proved] The degree-graded mass of the base-free design Mobius is 1 - 2t exactly. Degree is a monoid homomorphism M* -> N with finite fibres because 1 is the only constant in M*, which holds for F = {0,1} and for no design carrying a digit at least 2, where a constant c >= 2 makes the degree-zero fibre {c^k} infinite; pushing 1_(S*) * nu* = delta along it with 2^d polynomials of degree d gives A(t)/(1 - 2t) = 1, so the graded sums are 1, -2, 0, 0, ... and sum over deg P < L of nu*(P) = -1 for every L >= 2. The base-q design Mertens function is therefore pinned to -1 at every level boundary L >= 2 inside the carry-free window, and reads 0 at L = 1. Witness: lab/carry-free-mobius verb sequence.
• [Proved] The carry-free window of a two-digit design is (q+1)^(L-1) < q^L. A 0/1 polynomial of degree d has P(q) <= 1 + q + ... + q^d <= (1+q)^d and degrees add over a product, so the maximum of P(q) over M* at degree below L is exactly (q+1)^(L-1), attained by (1+x)^(L-1); hence the least base at which every element of M* of degree below L stays under q^L is the least q with (q+1)^(L-1) < q^L, for every L. Verified by direct enumeration at L = 3..14, where it reads 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 7, 7, while the maximum coefficient binomial(L-1, floor((L-1)/2)), A001405, stays measured and not proved, reaching 24310 at degree 17 against the crude proved cap 2^(L-1). The windows are 4, 7, 9, 12 and 15 at bases 3, 4, 5, 6 and 7, and sum_(n <= q^L) nu_F(n) leaves -1 at exactly level 5, 8 and 10 at the three bases where the departure is reachable, one level past the window each time, on the carries 4^4 = 256 over 3^5 = 243, 5^7 = 78125 over 4^8 = 65536 and 6^9 = 10077696 over 5^10 = 9765625. The running maximum is blunter and survives past the window, departing at level 9 at base 3 and 11 at base 4 and nowhere to level 11 at base 5, which is why 1, 1, 2, 3, 4, 7, 15, 23, 45, 86 is shared by base 4 and base 5. Witness: lab/carry-free-mobius verb lemma.
• [Proved] The design zeta has an explicit zero free half plane, and it closes the census right of the abscissa. Let a_min be the least nonzero digit of F, hence the least element of S_F, every element of two digits or more exceeding q. If a real sigma > alpha satisfies a_min^sigma zeta_F(sigma) < 2 then zeta_F has no zero in Re s >= sigma: the coefficients are nonnegative and the series converges for sigma > alpha, so for Re s = sigma' >= sigma one has abs(a_min^s zeta_F(s) - 1) = abs(sum_(n in S_F, n > a_min) (n/a_min)^(-s)) <= sum_(n > a_min) (n/a_min)^(-sigma) = a_min^sigma zeta_F(sigma) - 1 < 1. The hypothesis sigma > alpha is load bearing and the test is one real evaluation carrying the ladder's own error bound. On the grid alpha + 0.05 n the edge sigma_1 reads 0.5 at base 4 {1} to 1.75 at the three full digit sets over twenty-four designs, with a_min^sigma zeta_F(sigma) in [1.8635, 1.9995] and largest sigma_1 - alpha equal to 0.95, so a census of the zeros right of alpha needs no hand chosen right edge and the alpha + 3.02 strip of the locus sweep is three times wider than the zeros need. Witness: lab/transport-census verb census.
• [Verified] The transport census: every proper design censused carries zeros right of its abscissa, the full digit set alone carries none, and each rightmost is certified by a winding box. On the box alpha + 1e-6 < Re s < sigma_1, where the cofactor Z = zeta_F(s)(1 - k q^(-s)) is analytic and its zeros right of alpha are exactly those of zeta_F, the transfer failing only at residue null poles which sit on the line Re s = alpha, the argument principle counts 157 zeros right of alpha below Im s = 40 over twenty-three designs, all 157 located, plus 2 at base 50 missing one digit below Im s = 4. The count is exact on the box and a lower bound for the half plane, since the sliver alpha < Re s <= alpha + 1e-6, the band 0 < Im s < 0.02, everything above the census height and the conjugate half plane are uncounted. Twenty-one of the twenty-four designs carry such a zero; the three that do not are the base 2, 3 and 4 full digit sets, whose windings read -1.97e-33, 1.73e-33 and 1.53e-33. Every rightmost carries the height it is read below, because the teeth of the level zero comb drift right with the pole index: base 20 missing one digit reads 1.000285484146 at Im s = 2.0988, 1.000549674321 at 4.1971 and 1.002685494779 at 14.6920. Below Im s = 40 the rightmost real parts run 0.441505537191 at base 5 {0,1} to 1.002685494780 at base 20 missing one digit, each certified by a winding 1 box on zeta_F of half width 5e-5 in Re s and in Im s whose sampled contour minimum, 1.2e-4 to 6.1e-3, beats the engine's error bound by at least eight orders of magnitude and whose distance to the pole lattice s_(i,j) = alpha - i + 2 pi i j/log q is at least 0.00517845, four hundred box half widths. The two published boxes of lab/mrly-pairing reproduce at their own edges, winding 1 and 1 with contour minima 8.298e-4 and 6.865e-4, and its control rectangle returns winding 0. Witness: lab/transport-census verb census.
• [Proved] A certified zero right of the abscissa refutes every square-root-shaped bound for the design's own Mobius, and the digit 1 is the hypothesis that bites. Let 1 in F, let rho be a zero of zeta_F certified by a winding 1 box with left edge x_0 > alpha containing no pole, and let nu_F be the Dirichlet inverse of 1_(S_F). The transport theorem gives sigma_c(N_F) >= Re rho >= x_0 > alpha, so sum_(n <= x) nu_F(n) is not O(x^(x_0 - eps)) for any eps > 0; since A_F(x) has exponent alpha the square-root exponent is alpha/2 <= alpha < x_0, so the design's own Mobius satisfies no square-root-shaped bound and misses even the trivial O(x^(alpha - eps)), the first inequality failing to be strict only at the two designs with alpha = 0, where A_F(x) grows like log x. Nineteen of the twenty-four designs censused meet all three hypotheses and get a bound, and seventeen of the twenty-two the locus and family sweeps censused, the bounds running theta(nu_F) >= 0.4414555 at base 5 {0,1} to theta(nu_F) >= 1.0026354 at base 20 missing one digit. Four designs have Re rho > 1, so their own Mobius outruns the count of all integers below x: base 10 missing two digits, base 10 missing the digit 9, base 20 missing one digit and base 50 missing one digit, at k/q = 0.8, 0.9, 0.95, 0.98 and alpha = 0.9030900, 0.9542425, 0.9828779, 0.9948357; only the SIGN of Re rho - 1 is read and never its size, three of the four being censused to Im s = 40 and base 50 to Im s = 4. The k/q reading dies on its control, base 5 {0,1,2,3} at the same k/q = 0.8 with rightmost 0.989748105861. Two designs carry a zero right of alpha and no bound: base 4 {2,3} and base 4 {0,2,3} omit the digit 1, so 1 is outside S_F, the indicator vanishes there and nu_F does not exist. Witness: lab/transport-census verb law.

The design Mobius meter

• [Verified] The design Mobius meter oscillates at the zeta ordinates and not at the design's pole lattice. Read M_F(x)/x^(alpha/2) uniformly in log x, Hann-windowed, against a local-median floor and a null of rigid shifts of each candidate list: at base 10 with the digit 9 missing all six strongest peaks sit within one bin of a nontrivial zeta zero, offsets 0.068 to 0.216, with the full-set control at the same depth reading ten of ten, offsets 0.018 to 0.196. Thirteen zeta ordinates are reachable in the band 4 < gamma < 60, so a peak lands within one bin of one by chance with probability 0.159 and six of six is P = 1.6e-5. The pole lattice 2 pi j/log q scores -0.592, -0.640, -0.734 at base 3 {0,1}, base 3 {0,2} and base 5 {0,1}, below its own null, while the counting function over the identical elements scores 3.602, 3.764 and 3.973, so the pipeline would have seen a lattice and there is none. Witness: lab/design-meter verb spectrum, lab/mobius-designs, A084237.
• [Proved] The identity M_F(x) = sum_(n <= x) mu(n) A_F(n)/n + R_F(x) defines R_F at every base and digit set, and the echo's size splits at alpha = 1/2. Partial summation gives sum_(n <= x) mu(n) A_F(n)/n = A_F(x) H(x) - sum_(m in S_F, m <= x) H(m-1) with H(y) = sum_(n <= y) mu(n)/n, which is O(y^(-1/2 + eps)) under RH, so the echo is O(x^(alpha - 1/2 + eps)) when alpha > 1/2, while for alpha < 1/2 the second sum converges absolutely and the echo tends to the constant sum_n mu(n) A_F(n)/n, which is nonzero: base 16 {0,1} reads -0.0937, -0.1330, -0.1242, -0.1051, -0.1099 at 10^3 to 10^7 against x^(alpha - 1/2) falling 0.1778 to 0.0178, and base 10 {0,1} reads -0.0500 at 10^7 against 0.0405. Against the square-root bar x^(alpha/2) the echo dies at x^(-min(alpha, 1 - alpha)/2), equal to 1 only at alpha = 1, so the zeta zeros neither obstruct nor help the square-root conjecture, which is a statement about R_F alone. Verified separately at two designs: at base 10 missing 9 the echo carries six of six top peaks at zeta zeros and the residual none of the two it has, the echo being 0.1342 of the meter in root mean square against 0.6476, and the echo's share of the meter falls 0.356028, 0.242495, 0.207229 there and 0.208549, 0.099001, 0.047902 at base 3 {0,1}, share over prediction reading 1.0000, 0.7387, 0.6846 and 1.0000, 0.9219, 0.8663, each design decaying at least as fast as its own rate. Witness: lab/design-meter verb spectrum, mobius.md.

The zeros of the design zeta

• [Verified] This specific infinite design zeta has zeros in its own half-plane of absolute convergence, which the integers forbid, and the claim is the object and not the principle, since the positive-term Dirichlet series 1 + 2^(-s) has abscissa of absolute convergence -infinity and zeros at (2m+1) pi i/log 2. The census counts zeros of the Lyndon cofactor Z(s) = zeta_F(s)(1 - k q^(-s)), analytic on Re s > alpha - 1, so it needs no pole-free strip and leaves no sliver against the pole line, on the single strip alpha - 0.92 < Re s < alpha + 3.02, 0.02 < Im s < 60, split at Re s = alpha exactly. Base 3 F = {0,1} at alpha = log_3 2 carries 3 zeros right of the abscissa and 20 left of it inside that strip; base 10 missing the digit 9 at alpha = log_10 9 carries 13 right and 25 left; base 3 F = {0,2} carries 3 right, in the same three boxes as {0,1}. The largest surviving phase step on any census contour is 0.9896 and the largest propagated bound met at any census evaluation is 9.99e-11, both printed beside every count. The base 2 full digit set is the control on both sides and each side names its object: zeros of zeta in alpha + 0.02 < Re s < alpha + 3.02 count 0, which is what the Euler product forbids, computed and not quoted; zeros of zeta in alpha - 0.98 < Re s < alpha - 0.02 count 13, the first thirteen below Im s = 60; and the teeth of the cofactor 1 - 2 q^(-s), which sit exactly ON Re s = alpha and are not zeros of zeta, bring the one-strip count to 19 = 13 + 6 with 6 = floor(60 log 2/2 pi). Right of the abscissa no continuation is used, since the positive series converges absolutely there and the ladder only rearranges it. The count is resolved and not certified: the largest surviving phase step is printed and nothing bounds zeta_F'/zeta_F on the contour, so a zero pair closer than the surviving spacing would stay invisible. Witness: lab/design-zeta.
• [Proved] Scaled digit columns share a zero set exactly: for a positive integer a with a max F <= q - 1, so that aF stays inside {0..q-1}, the carry-free bijection m -> a m gives zeta_(aF)(s) = a^(-s) zeta_F(s), an exponential factor with no zeros and no poles, so zeta_(aF) and zeta_F have the same zeros and residues in the ratio a^(-s_(m,j)), and the proof uses 0 in F nowhere. Base 3 {0,2} against {0,1} agrees to 5.6e-43 at three points, and on what was censused, the strip alpha < Re s < alpha + 3.02, 0.02 < Im s < 60, the two censuses coincide box for box: winding one in Im [22.01, 24.01], in Im [28.01, 30.01] and in Im [56.00, 58.00] for both, and zero in every other box. Left of the abscissa {0,2} is not censused and is inferred from the theorem. On the Mobius side the same bijection twists the meter by a sign, so the transfer is exact on both faces and trivial on one of them. Witness: lab/design-zeta, mobius.md.
• [Proved] The Euler-product bridge between the two faces of RH is absent on a design: zeta M = 1 on the full set, S_F is not multiplicatively closed for any proper F, and zeta_F M_F is not 1, so no known route runs from a zero of zeta_F to theta(F) and the zero census carries no bound on the square-root conjecture. What survives is not the zeros but the position product: G_L pairs against a = 1 and a = mu alike and the arithmetic sits entirely in the kernel. Coons 2010 Theorem 2.3 rules out the automatic-continuation route to M_F and nothing wider. Witness: mobius.md, lab/design-zeta, lab/mrly-euler, REFS.md.
• [Proved] The zeros of the design zeta are read off one analytic function and their positions near the pole lattice are forced by the residues. The Lyndon cofactor Z(s) = zeta_F(s)(1 - k q^(-s)) is analytic on Re s > alpha - 1, since 1 - k q^(-s) cancels exactly the m = 0 line of the digit recursion's poles and no other; the poles of Z are those s_(m,j) = alpha - m + 2 pi i j/log q with m >= 1 at which zeta_F has a nonvanishing residue, the nearest line to that half-plane being Re s = alpha - 1 with residue -s_(1,j) gamma_1 r_j/k, and on a full digit set Z has no pole at all, being zeta(s)(1 - q^(1-s)), entire. One peel level gives Z(s) = E_1(s) + sum_(l >= 1) binom(-s,l) q^(-s-l) gamma_l zeta_F(s+l) with E_1(s) = sum_(a in F, a != 0) a^(-s) and gamma_l = sum_(a in F) a^l, checked against brute-force digit summation to 1.6e-14 with and without 0 in F, and Z(s) a_min^s -> 1 to the right. A zero of zeta_F right of alpha - 1 is always a zero of Z; conversely a zero of Z is a zero of zeta_F except at a pole s_(0,j) with r_j = 0, where Z vanishes and zeta_F is regular. At s_(0,j) one has k q^(-s_(0,j)) = 1 exactly for every j, so with u = s - s_(0,j) the periodic factor is 1 - q^(-u) with no j dependence and zeta_F(s) = Z(s)(1/(L u) + 1/2 + L u/12 - L^3 u^3/720 + ...), L = log q, giving residue Z_0/L, regular part Z_1/L + Z_0/2 and its derivative Z_2/L + Z_1/2 + Z_0 L/12 from the Taylor coefficients of Z alone, on a disc of radius at least 1 and exactly 1 when r_j does not vanish. Witness: lab/zeta-locus, lab/design-zeta, lab/burnol-residue.
• [Verified] The zeros of the design zeta near the abscissa are a residue comb whose tooth position the residue and the regular part predict. A zero near the pole s_(0,j) solves u(R_j + R'_j u + ...) = -r_j, first order u_1 = -r_j/R_j and second order the near root of R'_j u^2 + R_j u + r_j = 0, both built from the Laurent data with nothing fitted. Over 20 designs to Im s = 40 (every scaling class at q = 3 and q = 4, two at q = 5, base 9 {0,1,2}, base 16 {0,1,2,3}, base 10 missing 9, base 2 full set) one assignment radius 0.45, fixed by the discs not overlapping and not by the tooth law so that every count is conditional on it, serves both the count and the tooth, discs never overlapping since the smallest period in the sweep is 2.2662: the argument principle on that circle gives 164 poles carrying one zero of Z, 40 none and 8 two, of which 21 are the residue-null pole centres of the three full-set columns and are zeros of Z that are not zeros of zeta_F, leaving 143 poles with one zero of zeta_F, 61 with none and 8 with two. All 151 poles carrying a zero have their zeros located by a polar grid inside that same disc and not by the prediction, so no tooth is selected by the law it tests and no pole carrying a zero is left without one. Comparing prediction to tooth afterwards, miss2/miss1 has median 0.1637 with miss2 < miss1 at 147 of the 151, and the accuracy is conditional on the tooth being close: the 43 teeth at abs(u) < 0.1 have largest first-order miss 0.01446 and largest second-order miss 0.00164, the 84 at abs(u) < 0.2 have 0.10815 and 0.01526, while the 31 at abs(u) >= 0.3 reach 1.64614 and the prediction says nothing. The densest column is the sharpest: base 10 missing 9 at k/q = 0.9 locates 15 teeth to a largest first-order miss of 0.013602 and a median of 0.000841. Witness: lab/zeta-locus.
• [Verified] The critical line is the second family of the full digit set. On a full digit set zeta_F is zeta, whose only pole is s = 1 = alpha, so it is regular at every s_(0,j) with j != 0 and the residue there vanishes as a one-line consequence rather than a measurement; the machinery reads those residues as 1e-26 to 1e-33, which is a control of the engine, and the comb is empty. The winding of the cofactor over alpha - 0.92 < Re s < alpha + 3.02, 0.02 < Im s < 40 then splits exactly as six zeros of zeta plus floor(40 log q/2 pi) cofactor-only teeth, those teeth being the zeros of 1 - q^(1-s) on Re s = 1 by exact arithmetic: 10 = 6 + 4 at q = 2, 12 = 6 + 6 at q = 3 and 14 = 6 + 8 at q = 4. The six survivors read Re s = 0.5 at Im s = 14.1347251417, 21.0220396388, 25.0108575801, 30.4248761259, 32.9350615877, 37.5861781588 at all three bases, and the three columns share that zero set to 1e-26 because they are one arithmetic object. On a design the same split leaves a second family that is not a line at alpha/2: its real parts run -0.273079611 to 0.391038600 over the 7 zeros below Im 40 at base 3 {0,1} against alpha/2 = 0.3154648768, -0.30495894 to 0.28101268 over 6 zeros at base 4 {0,1} against 0.25, and 0.060261843 to 0.97363028 over 5 zeros at base 16 {0,1,2,3} against 0.25, the spread being the witness and no per-design mean claimed. Witness: lab/zeta-locus.
• [Proved] The second family of the design zeta does not depend on which comb is stripped, and the next pole line's comb is computed from the first one's residues. For m >= 1 the cofactor Z_m(s) = zeta_F(s) prod_(i <= m)(1 - k q^(-(s+i))) has exactly the zeros of Z(s) = zeta_F(s)(1 - k q^(-s)) inside alpha - 1 < Re s < alpha + 3.02, since each extra factor vanishes only on Re s = alpha - i for i >= 1, so the survivors of the assignment are one set under every comb. What Z_m adds is the level-i comb, and its Laurent data is forced by the level-zero data: Z(s) = E_1(s) + sum_(l >= 1) binom(-s,l) q^(-s-l) gamma_l zeta_F(s+l) is singular at s_(1,j) = alpha - 1 + 2 pi i j/log q through its l = 1 term alone, and with q^(-s_(1,j)-1) = q^(-s_(0,j)) = 1/k and 1 - k q^(-s_(1,j)) = 1 - q the residue of zeta_F there is r_(1,j) = s_(1,j) gamma_1 r_(0,j)/(k(q-1)), so the level-one comb is empty wherever the level-zero comb is, and at the full digit set s_(1,0) = alpha - 1 = 0 makes it vanish, which is zeta having no pole at s = 0; the lab prints abs r_(1,0) = 0.0 with the null flag set and abs r_(1,1) = 8.89623e-29 at the base 2 full set. Witness: lab/zeta-family, lab/zeta-locus, lab/design-zeta.
• [Verified] The second family of the design zeta, split out and censused over twenty-two designs at a stated assignment radius, with no gap at that radius on any design. Stripping the level-zero and level-one combs at rho = 0.45, a constant fixed only by the pole discs not overlapping and not by the tooth law, which is accurate only inside abs(u) < 0.2, the twenty designs of the locus sweep plus base 5 {0,1,2,3} and base 10 missing two digits give 377 zeros wound by the argument principle, 351 located, 171 teeth of which 9 are level-one teeth, 19 cofactor-only zeros at null-residue poles and 161 second-family zeros, each design censused to its own printed height, 40 except the four base 3 designs at 42.894, base 9 {0,1,2} at 41.464 and base 10 missing two at 25.923. There is no gap at rho on a design: the distance from a second-family zero to the nearest live pole has minimum 0.45510938 at base 4 {2,3}, 0.45909168 at base 3 {0,1}, 0.48696667 at base 4 {0,1,2} and 0.50481072 at base 4 {1,3}, with base 4 {2,3} putting five of its eight inside 0.45 < abs(u) < 0.6, so every count is conditional on rho and falls as rho rises, N_2 reading 8, 7, 13, 9, 14 at rho = 0.45 against 7, 6, 12, 8, 7 at rho = 0.6 on base 3 {0,1} and the four base 4 two-digit designs. The full digit set is where the gap exists: at base 2 the nearest live pole to a second-family zero is 14.143566 away and no radius below 0.9 moves any count. Where the located count falls short of the winding, base 4 {2,3} at 12 of 18 being the worst, N_2 is a lower bound. Witness: lab/zeta-family verb tests.
• [Verified] Seventeen designs carry a lower bound on the Mertens exponent of their own Mobius, and the strongest bound is radius-robust. Nineteen of the twenty-two designs have a censused zero of zeta_F strictly right of alpha, twelve of them in the second family, and at the seventeen of them whose digit set contains 1, so that nu_F exists, the transport theorem gives that sum_(n <= x) nu_F(n) is not O(x^(Re rho - eps)); base 4 {2,3} and {0,2,3} omit the digit 1 and carry a zero but no nu_F. Base 10 missing two digits has a zero at 1.00151438765 + 2.77402670058 i against alpha = 0.903089987, a second base-10 column where the design's own Mobius has a Mertens exponent above 1 and so above x itself; that zero is a level-zero tooth at abs(u) = 0.1083 of the j = 1 pole, deep inside every assignment radius tested, so the bound does not depend on where the comb is cut. Base 4 {1,2} has a second-family zero at 0.940012431696 + 13.0678968771 i against alpha = 1/2, an exponent of 0.94 against a design mass exponent of 0.5, and base 3 {0,1} reads 0.720787601477 at Im 28.6056765649 against alpha = 0.630929754. Witness: lab/zeta-family verb tests, lab/mrly-pairing verb inverse.
• [Verified] What converges as a design fills is the ordinate set and not the real part. Against the derived null of a quarter of the mean gap between consecutive zeta ordinates in the range, the exact expectation for an equally spaced ordinate set of the same density and conservative for one with gap variance, the mean distance from a second-family ordinate to the nearest zeta ordinate divided by that null falls monotonically in alpha: 2.0495374 at base 5 {0,1} with alpha = 0.430676558, 1.8953371 at base 4 {0,1} with 0.5, 0.75419266 at base 3 {0,1} with 0.630929754, 0.51648744 at base 4 {0,1,2} with 0.792481250, 0.32356636 at base 5 {0,1,2,3} with 0.861353116, 0.090501352 at base 10 missing two with 0.903089987 and 1.0429899e-23 at the base 2 full set. The base and k confounds are dead: the fall is monotone at fixed base, 2.0495374 to 0.32356636 inside base 5 and 1.8953371 to 0.51648744 inside base 4, and at fixed k = 2 across bases, 2.0495374, 1.8953371, 0.75419266, 1.0429899e-23 at alpha = 0.430676558, 0.5, 0.630929754, 1; the nulls move only 1.0425839 to 1.3595166 across the ladder while the raw mean distance falls 2.4032315 to 0.12303809, so the denominator does not drive it. Over the same designs mean abs(Re s - 1/2) reads 0.36482392, 0.39426128, 0.3901396, 0.25540269, 0.31452367, 0.20473972 and 2.4065966e-23 and does not fall monotonically, so at alpha = 0.903 the heights are pinned to 2.3 percent of the mean gap while the real parts are still 0.20 off 1/2. alpha is a trend and not a function: the four base 4 two-digit designs at one alpha = 1/2 spread 0.79050661 to 2.8404536. The matching is nearest-ordinate and not injective, 3 distinct ordinates for 4 design zeros at base 10 missing two. Witness: lab/zeta-family verb limit.
• [Proved] The ordinate shadow is a first-order perturbation and its constant-free form is a Newton step from the zeta zero. The discrete position identity 1_(D_L)(n) = q^(-L) sum_(a mod q^L) G_L(a/q^L) e(-n a/q^L) on 0 <= n < q^L gives zeta_(F,L)(s) = q^(-L) sum_(a mod q^L) G_L(a/q^L) S_L(s, a/q^L) with S_L(s,x) = sum_(1 <= n < q^L) e(-nx) n^(-s), reproduced from the transform to 8.326e-40 at L = 2 on ten designs, and since G_L(0) = k^L the a = 0 fibre carries the weight (k/q)^L exactly against the partial sum of zeta to q^L, with no arc and no limit. That identity splits the level-L polynomial against a TRUNCATED zeta while the object is the continued zeta_F against the full zeta, and (k/q)^L falls to 0 with L while both series tend to 1 on the right, so no level is forced and c = k/q is the L = 1 reading and a definition. For any constant c the split zeta_F = c zeta + E_F gives E_F(rho_0) = zeta_F(rho_0) at a zero rho_0 of zeta, an identity carrying no information about c, and a first-order zero of zeta_F at rho_0 - zeta_F(rho_0)/(c zeta'(rho_0)); reading c zeta'(rho_0) as zeta_F'(rho_0) removes the constant and gives rho_0 - zeta_F(rho_0)/zeta_F'(rho_0), Taylor at a simple zero of zeta_F. The offset is one complex number, so at the zeros this law pairs the ordinate offset and the real-part offset are one quantity. The continuous form, the mass of G_L on abs(t) < 1/(2 q^L), is the exact sinc sum 1/q^L + sum_(n in D_L, n > 0) sin(pi n/q^L)/(pi n) and equals kappa_L(F) (k/q)^L with kappa_L running 0.6015221 to 0.96774464 over the ladder at L = 1, 2, 3, so it adds no constant the fibre does not give. Witness: lab/zeta-shadow verb mass, lab/mrly-euler verb position.
• [Verified] The constant-free first-order step predicts the design zero attached to each zeta zero, and it sharpens as the offset shrinks. Over nine designs at twelve zeta zeros to Im s = 56.4462476971, six to Im s = 37.5861781588 at the two densest so the rungs do not share one height, both predictions are computed from zeta_F(rho_0), zeta_F'(rho_0), zeta'(rho_0) and the digit density alone and the zero is located afterwards by Newton from rho_0, accepted only at abs(zeta_F) < 1e-16, within 1.5 of rho_0 and 0.02 clear of the pole lattice, largest ladder bound 9.001e-23. The step's median ratio reads 1.3843088, 1.284225, 1.2481449, 1.2060106, 1.2042502, 0.89075541, 1.0195598, 1.005076, 0.99741809 at alpha = 0.430676558, 0.5, 0.630929754, 0.792481250, 0.861353116, 0.903089987, 0.954242509, 0.982877878, 0.994835739, with largest abs(ratio - 1) 0.14041 at base 20 missing one digit and 0.01734 at base 50 missing one digit, bands [0.94875, 1.14041] and [0.98266, 1.01144]; pooled over the ladder that largest deviation runs 0.01734, 0.0508884, 0.193158, 0.83912, 3.32327 over the buckets abs off < 0.05, < 0.1, < 0.2, < 0.4 and above, on 7, 4, 14, 18, 44 zeros. The L = 1 reading c = k/q is the looser column, median ratio 1.4129353, 1.2842149, 1.0955991, 1.1806066, 1.1372453, 1.276577, 1.1347487, 1.0320127, 1.0507079 with largest abs(ratio - 1) 0.24964 and 0.0821168 at the two dense rungs, five times looser than the step at base 50, and the coupling zeta_F'(rho_0)/zeta'(rho_0) does not select it either, median abs(coupling - k/q) reading 0.24057225 and 0.08291158 there against median abs(coupling - 1) 0.27619434 and 0.079335871, a flip between the two rungs while the candidates differ only by 0.05 and 0.02. Nine zeros at the three sparsest designs have no located zero inside the trust region, predicted offsets 0.95618855 to 3.0967393, so those rungs' medians are conditioned on Newton succeeding. The base 2 full set is the exact control, abs(zeta_F(rho_0)) between 1.85e-34 and 1.329e-25 at all twelve zeros, so E_F = 0 and both offsets are 0. Witness: lab/zeta-shadow verb predict.
• [Verified] The paired shadow offset carries its exponent in the missing-digit density rather than in 1 - alpha, and two new rungs sample the interval between alpha = 0.954 and 1. The median paired offset divided by m/q = 1 - k/q reads 1.6463532, 1.2495026, 1.8345578, 1.5731321, 2.2102406, 1.6634381, 2.2424916, 1.8779239, 1.051349 across the nine rungs and divided by 1 - alpha reads 1.7350628, 1.2495026, 1.6569184, 1.8951686, 3.1883019, 3.432954, 4.9008186, 5.4839111, 4.0716338; a least squares in the logs, a fit and not a theorem, gives (m/q)^1.04544 at R2 0.957842 against (1-alpha)^0.71691 at R2 0.944011, the first column spanning 2.13297 and the second 4.38888, so m/q carries the exponent by a factor of 2.05764 inside the 4.28797 that (1-alpha)/(m/q) itself spans over this ladder, which is the whole discrimination the two normalisations admit here. The new rungs are base 20 missing its top digit at alpha = 0.9828778777 and base 50 missing its top digit at alpha = 0.9948357391, all six zeros located at each, median abs(E_F(rho_0)) 0.11830158 and 0.028066806 and median offset 0.093896196 and 0.021026979, so the PAIRED offset falls fast across that interval; this bounds no maximum over the whole second family and touches no jump clause, since the pairing selects zeros for closeness to a zeta zero and censuses nothing. Read in the form of the family row, the mean distance from a located design ordinate to the nearest zeta ordinate over a quarter of the mean gap between consecutive zeta ordinates in the range gives 0.81218635, 0.57141859, 0.50488757, 0.37447728, 0.20954319, 0.29197634, 0.14794812, 0.052888241, 0.011098646 and 0 at the full set; the pairing is zeta-zero-first where the family row's is design-zero-first, so this is a parallel ladder and not that row recomputed. Witness: lab/zeta-shadow verb rungs.
• [Proved] A positive Rouche margin proves exactly one zero of the design zeta in a disc about a pole, with every input bounded from the digit recursion itself. Write Z(s_0+u) = P(u) + T(u) at a pole s_0 = s_(0,j) with nonvanishing residue, where P(u) = (1 - q^(-u)) D_(P-1)(s_0+u) + E_P(s_0+u) is entire with Taylor coefficients the exact finite sums sum_n n^(-s_0)(-log n)^m/m! convolved against those of 1 - e^(-L u), and T is the l >= 1 part of the ladder numerator, bounded on abs(u) <= R_2 by B_T = sum_(l >= 1) binom(abs(s_0)+R_2+l-1, l) q^(-sigma-l) gamma_l G(sigma+l) at sigma = Re s_0 - R_2 with G the peeled majorant. That l sum is closed by a majorant ratio and not by an observed one, the term ratio itself not being monotone: since gamma_(l+1)/gamma_l <= a_max and G(sigma+l+1)/G(sigma+l) <= q^(-(P-1)) because every string in the pools is at least q^(P-1), the term ratio is at most R_l = ((abs(s_0)+R_2+l)/(l+1)) a_max q^(-P), which decreases in l once abs(s_0)+R_2 >= 1 and is below a_max q^(-P) otherwise, so stopping at the first l with R_l < 1 and adding term_l R_l/(1-R_l) is a proof. Then abs(Z_n) <= B_T/R_2^n for n >= 2 beyond the explicit part, so on abs(u) = rho one has abs(Z - (Z_0 + Z_1 u)) <= sum_(m >= 2) abs(P_m) rho^m + B_T tau^2/(1-tau) with tau = rho/R_2, while abs(Z_0 + Z_1 u) >= abs(Z_1) rho - abs(Z_0); when the first is strictly less than the second the linear model and Z have the same zero count in abs(u) < rho by Rouche, and that count is one because abs(Z_0/Z_1) < rho follows from the same inequality. Since the residue does not vanish, Z(s_0) != 0 and the zero is a zero of zeta_F. No step uses a differenced quantity: Z_0 is the ladder value with its propagated bound and Z_1 is the first Fourier mode of T on a circle of radius R < R_2 with N samples, whose aliasing is at most (B_T/R_2)(R/R_2)^N/(1-(R/R_2)^N), plus an exact p_1. The peel depth P and the radii rho and R_2 are free parameters of the proof. Witness: lab/zeta-locus, lab/design-zeta.
• [Verified] The residue comb carries exactly one zero of the design zeta at eleven certified poles, the peel depth is the lever that decides which, and the certificate fails at every pole carrying none or two. Running the Rouche margin with the peel depth raised at each pole until the certificate fires or the string pool caps, over 106 poles at base 3, base 5, base 9, base 16 and base 10 missing 9 to Im s = 40 inside a fifteen minute budget, gives 11 certified, 60 failed, 7 residue-null and excluded because there the model's zero is the pole centre, a zero of the cofactor that is not a zero of zeta_F, and 28 skipped when a design spent its budget. The certified eleven, with depth, margin and the radius the proof used: base 3 {0,1} j = 2 at P = 7, 0.13418242, rho = 0.205; j = 5 at P = 7, 0.028140545, rho = 0.16; j = 7 at P = 9, 0.00082974181, rho = 0.175; base 5 {0,1} j = 4 at P = 7, 0.15035818, rho = 0.2775; j = 5 at P = 7, 0.12269904, rho = 0.295; base 9 {0,1,2} j = 5 at P = 5, 0.038456894, rho = 0.26; and base 10 missing 9 at j = 1, 2, 3, 4, 7, all at P = 3, margins 0.047105507, 0.030062806, 0.045802462, 0.043508323 and 0.046292701 at radii 0.1275, 0.105, 0.1025, 0.09, 0.0725, each on 24 contour samples. Every certified disc agrees with the argument principle count of one and none disagrees; of the 19 poles carrying zero or two zeros in abs(u) < 0.45 that the budget evaluated none is certified, the two double poles reached, base 3 {1,2} j = 3 and j = 5, both failing, while base 5 {1,2} j = 7 and base 10 j = 15 were skipped for budget. Base 10 is not closed by any sharper majorant but by peeling: at the automatic depth P = 2 its B_T runs 1.08 at j = 1 to 38.1 at j = 15, and at P = 3 it runs 0.2096 to 1.2010 over the eight poles reached, five of which certify. Proximity of the tooth is no threshold, the certified abs(Z_0/Z_1) running 0.0282669 to 0.149708 and base 3 {0,1} j = 7 at 0.104443 failing at P = 7 and certifying at P = 9. The margins are evaluated in high precision and not in ball arithmetic, which is the one step between this row and Proved. Witness: lab/zeta-locus.

OPEN

Coprimality density

• [Conjecture] The visible density inside a design's q-periodic pattern is exactly delta: 4/pi^2 = 0.405284734569 for the base-2 gasket pattern, 21/(4 pi^2) = 0.531936214122 for the carpet, 19/(26 zeta(3)) = 0.607932310732 for the sponge, by coprime tuples splitting evenly over the nonzero residue classes ((2/3)(6/pi^2), (7/8)(6/pi^2), 19 of 26 classes over 1/zeta(3)), with worst Mobius-count error 6.05e-07 at N = 10^6; designs differing only in the all-zero corner have identical visible density on 8579 pairs, since that class holds no visible points.
• [Conjecture] Exhaustive endpoints at bases 2..6 and D = 2, 3 agree with the predicted delta with no inferable rate: base-6 code 34376528265 reads 0.454413 against 0.455945 at n = 8, and the base-5 D = 3 pair is the worst case at 8.3e-03 and 1.5e-02.
• [Conjecture] The central-slice peel ratio tends to (sqrt(33) - 5)/8 = 0.0930703308, measured 0.093070331, forced from the shared (9, -12) recurrence of the peeled streams, which itself stays Conjecture. Witness: lab/slice-coprimality.

Walk dimension

• [Conjecture] Fixed mass does not fix the drum either: all nine fill-8 tiles recomputed at n = 243 and n = 729 under both boundary conventions give centre 1.2111, edge-midpoint 0.9661, corner 0.9634 (shipped convention 1.2093 / 0.9666 / 0.9633) at full held fraction, so the 0.25 spread is not a dropped-component artifact and "mass falls faster than stiffness" fails at exactly fixed mass.

Flake band gap

• [Conjecture] The lower edge closes at a fitted c * 8^(-L) with c near 12.9868 over ten levels, (2 - lo) * 8^L reaching 12.984807 and 12.986289 at L = 7, 8, a fit with no mechanism. Witness: lab/flake-band-gap.

Odd-base slice grammar

• [Conjecture] No definition of a "blow-up of 4" for the slice exists in this tree, so the phrase carries no claim.
• [Conjecture] The two-tile grammar closes at ten odd bases with a 2 x 2 substitution matrix rational in b within each class of b mod 4.
• [Conjecture] That parity forces structurally different cells into the middle layer in each residue class, which is the mechanism of the mod-4 split.

Flat carpet stack

• [Conjecture] The rendered diagonal-to-background ratio reads 2.1189 at N = 55 walking to 2.0000252 at N = 10^6, and the rendered grey ratio is 16/9 because ink is 17.
• [Conjecture] A primitive integer line alpha u + beta v = gamma is a ray iff alpha and beta are both odd, and the crosshairs at u = a/b carry 1/(4b), a factor b stronger than the diagonal family.
• [Conjecture] Pi survives only in the census and the decay arithmetic: distinct rays are indexed by odd-denominator reduced fractions, counted by sum phi(b) ~ (2/pi^2) B^2 through the odd-prime Euler product 8/pi^2, and the pairwise gcd sum obeys S2(N)/N -> pi^2 ln 2/(7 zeta(3)) = 0.8130217 (6 digits at N = 10^6 two ways), whence the tree constant lim L * Var = pi^2 ln 2/(14 zeta(3)) = 0.4065108521 and the carpet and net constant 0.2724570.
• [Conjecture] The sup-norm never fades: diagonal, crosshairs and the four brightest points (paper 9/14 at the inner-thirds crossings, grey exactly 170) hold their values forever while their width shrinks like 2/(N + 1) on an exactly triangular profile.
• [Conjecture] Chaining 27 8-bit blends and truncating each step shifts the rendered stack by six grey levels, the saved image's mean 61.9 against the true 67.8 and its brightest pixel 163 against 170; quote paper-coverage fractions, never absolute greys.
• [Conjecture] The 2D ray law transfers to the void flat stack at double contrast: void has no crosshairs (no single-wave terms) but carries the full odd/odd 2-adic diagonal web at twice the carpet's strength (offset a/b: (-1)^a/(2b^2); slope q/p through the origin: 1/(2pq)), with u = v and u + v = 1 solid ink, a black X on mid-grey, the negative of the carpet's paper X.

Diagonal slice stack

• [Conjecture] On rendered cut grams masked to the common hexagon, the 290 coprime layer pairs have Pearson mean -0.037 and range [-0.205, +0.147] with 85 of 290 beyond |0.05|, against a flat-stack coprime maximum of 0.017 on the same raster and exactly 0 in the continuum; the gcd echo survives with the (m, 3m) family topping the table at (17, 51) = +0.248; the two strongest coprime pairs, (5, 9) = -0.205 with the same residue mod 4 and (5, 7) = -0.204 with different residues, show the mod-4 alternation is not the mechanism; these half-mask values are superseded by the full-hexagon -0.142 and -0.085 below.
• [Conjecture] The xor pair's 14 + 14 stack is the flattest nontrivial field measured, carrying six eternal points at the permutations of (1/4, 1/4, 1), ink at all 28 layers for 105 and paper at all 28 for 150, by a one-line parity proof.
• [Conjecture] The doubling magnitude is 253/2160 = 0.11712963, fitting both branches to 1e-6. Witness: lab/hexagon-moire. Superseded: the constant is exactly 253/2160 by the phase-map integral, see the layer-pair row under Diagonal slice stack in SETTLED.
• [Conjecture] The Catalan statement M (mean ink - 1/2 - eps/2) -> G/8 holds only along N = 3 mod 4 (0.1144757884 at N = 55, 0.11448 at M = 28 against 0.11450); along N = 1 mod 4 the limit is G/8 - 1/8 (-0.0104828892 at N = 53). Witness: lab/hexagon-moire. Superseded: the statement is Proved at both residue classes by the summed ink law, see the Catalan row under Diagonal slice stack in SETTLED.

Dimension one

• [Proved] The tenth rung of the moment ladder moves the bottom edge to 0.4475978 (beta_0^(10) = 0.4475978134...), leaving the standing window (0.4475978, 0.6402122] with both edges unconditional; the eighth rung at 0.446717 is now only a table row. Witness: lab/dimension-one-ladder.
• [Conjecture] Occupancy is the one door left in the 1/2 wall: the Cauchy-Schwarz bound saturates at beta = 1/2 against the trivial ray count, but occupied rays number only 3^(0.5416 n) to 3^(0.5798 n) at the critical band against the trivial 3^n under the pinned threshold reading, n = 10..18 (the earlier band 0.543 to 0.557 does not reproduce), every occupied ray has exactly one coordinate divisible by 3, and Theorem R+ closes the entire window under Conjectures Z and O while every bootstrap from Z to O collapses to the trivial fixed point; minimal witnesses are not unique (four tied rays at n = 12, repaired by a least-multiplier tie-break) and prefix-newness is necessary but not sufficient, overcounting occupied rays by a stable 1.51x. Witness: gasket-ray-machine, lab/dimension-one-ladder.
• [Conjecture] Higher ray-mass moments make the Holder conversion strictly worse, capping the ray power-moment route at the second-moment edge 1/2: at n = 12 the 345318 occupied rays have S_1 = 523250, S_2 = 1374038, S_3 = 46380938, S_4 = 8145428822, max M = 232, and the Holder bound S_1 <= N^(1 - 1/r) S_r^(1/r) overshoots by factors 1.316, 3.380, 8.179 at r = 2, 3, 4, because Fibonacci-heavy shift rays dominate the high moments (phi > 3^(1/(2K)) at the critical half-scale); the ladder n = 8..12 lists occupied rays 3904, 12170, 37298, 113836, 345318 with max M 33, 54, 88, 143, 232.
• [Proved] Conjecture O has no content below alpha = 1/2 - the rays of height at most 3^(alpha n), occupied or not, number at most 3^(2 alpha n) under the threshold reading and 9 * 3^(2 alpha n) under the octave cut, so the box alone gives delta = 1 - 2 alpha with no occupancy input, and the whole conjecture lives in alpha in [1/2, 0.5533]. Witness: lab/occupancy-decay.
• [Proved] The first moment of occupancy is the window itself, so no proof of O may pass through it - with F(n, X) the non-fibre gasket points of primitive height at most X, Sum_{p > 3^(beta n)} N_n(p) <= (F(n, 3^((1-beta) n)) + 2^(n+1)) / beta at target zero, each such x carrying at most 1/beta primes above 3^(beta n); a first-moment bound at alpha > 0.3597878 moves the standing window and at alpha >= 0.5524022 closes it with no Conjecture Z, and the inequality holds with ratio 0.0846 to 0.1517 against the sieved prime sum at n = 10, 12, 14, beta = 0.45, 0.5, 0.6. Witness: lab/occupancy-decay.
• [Verified] Occupancy pays no exponent for the multiplicity, so O carries the full weight of the window and is no cheap half of Theorem R+ - F/A at alpha = 0.5533 reads 5.41, 5.20, 5.52, 5.64, 5.63, 5.86, 5.79, 6.08, 5.92 at n = 10..18 while log_3 F / n falls 0.7645 to 0.7201 against log_3 A / n inside [0.6109, 0.6345], the exponents converging at log(F/A)/(n log 3); only at fixed height do the shift rays split them, A(n, 3^5) = 384 .. 474 against F(n, 3^5) = 2728 .. 51694. Witness: lab/occupancy-decay.
• [Verified] The digit-congruence seed is measured out as a route to O - the proved bound A(n, X) <= 2 sigma_k X^2 + 2 sigma_k 3^k X + 2 X^2 3^(-k) + 3^k + 2 X for 3^k <= X collects every digit-class constraint, the mod-3 dichotomy being k = 1, but sigma_k = |R_k|/3^k falls only polynomially through k = 18, |R_k| = 73440, 206149, 580920, 1643545, 4663382, 13272515 at k = 13..18 with growth rising 2.794 to 2.8461 and k(1 - log_3 growth) inside [0.8418, 0.8628], so the route buys n^(-0.86) and no exponent; on the measured hypothesis M_2(k) = O(4^k) (M_2/4^k = 0.4098, 0.4077, 0.4071, 0.4029 at k = 13..16, still falling) Cauchy-Schwarz caps any congruence-only decay at c = 0.2618596, alpha = 0.575328, excluding neither 0.5533 nor 0.5524022, and no exponential floor is proved either way. Witness: lab/occupancy-decay.
• [Verified] Occupied non-fibre ray totals 1044840, 3151656, 9491964, 28545340 at n = 13, 14, 15, 16 from a second builder, two below the earlier totals at every level, exactly the two fibre rays. Witness: lab/occupancy-decay, lab/dimension-one-ladder.
• [Proved] The golden ceiling M_n(z) <= F(n+1) - 1 holds for every direction of the 13158-box at every level, promoted from an enumeration to n <= 40 - in the direction coordinate a multiplier word is a word over the increments {0, z_2, -z_1} summing to zero, its carry automaton has out-degree at most 2 with the branch states in one residue class mod 3, and the two successors of a branch state differ by q/3 for the unique q in {z_1, z_2, z_1+z_2} divisible by 3, so when no branch state has two branching successors (in particular when v_3(q) = 1) the state maximum obeys G(n) <= G(n-1) + G(n-2) and the ceiling follows; that settles 206 of the 218 occupied directions, 107 by v_3(q) = 1, three of the twelve left are shift rays closed by F(p+2) F(q+2) = F(p+q+3) - F(p+1) F(q+1), and nine carry rational Fibonacci certificates of denominators 18, 40, 381, 18, 2013, 18, 40, 2013, 34. The hypothesis z_1, z_2 >= 1 is load-bearing: on the fibre ray (0,1) two digits share the increment 0, a set-valued reading sees no branch state, and M_6(0,1) = 63 against F(7) - 1 = 12. Refutation attempt: ground truth rebuilt independently from the ray definition for 20 directions including all twelve hard ones, zero mismatch; all nine certificates re-verified in exact rational arithmetic with domination checked to n = 60; the box census, the renewal criterion, the (1,9) profile, the weight bound and the -1 path accounting all recomputed exact. Witness: gasket-ray-machine, lab/gasket-witness-weights.
• [Proved] The whole case list collapses into one algebraic inequality per direction. Weight the first returns of the direction automaton by phi^-1 a step: with g(c,m) the paths from a live state c to the start meeting it only at the end, u(c) = Sum_m g(c,m) phi^-m and U(z) = Sum u(c') over the start's successors other than itself, so Sum_{j>=2} f_j phi^-j = phi^-1 U; any pi > 0 with Sum_succ pi <= phi pi(c) at every live c != 0 and Sum_(c' != 0 succ 0) pi(c') <= phi^-2 pi(0) forces U(z) <= phi^-2 by a maximum principle on the truncated sums, and then M_n(z) <= F(n+1) - 1 at every n by renewal against the envelope phi^(m-2) <= F(m) <= phi^(m-1), with pi = u admissible as soon as U(z) <= phi^-2. It proves 45 directions no earlier case reached, the nine hand-tuned rational certificates and the 36 that rested on enumeration alone; 3 nmid z_1 z_2 gives M_n = 0 outright, settling 6566 box directions on residues against 3284 before; f_1 = 1 always and f_2 = 1 only at {a,b} = {1,3}, both from the increments. Refutation attempt, briefed to break it: the proof read line by line for convergence, normalisation, S = phi^-1 U and both envelopes; an independent Q(sqrt5) rebuild reproduced every count (218 occupied and 214 passing in the box with the four shift-ray failures, 647 and 644 outside it with three, 865 and 858 in total with seven, 57 distinct U values, the exact attainers); a 600 x 1800 box census with 3866 occupied directions, 4.5 times the shipped range, plus 19681 stressors including 52 in the open v_3(q) >= 2 ground, found failures only at shift rays; and every shipped solve is confirmed strictly positive and against both criterion inequalities, not merely against the linear system. Witness: gasket-ray-machine, lab/gasket-witness-weights.
• [Verified] The golden potential misses exactly the shift rays and, on every censused range, nothing else: U(1,3^j) = phi^-1 exactly because the shift mass grows at rate phi, U = phi^-2 only on the supergolden (1,12), (3,10), (4,9), and no direction of any range censused has U in the open interval (phi^-2, phi^-1) - the box, the six adversarial families, a 36037-direction lab sweep with high-v_3 stressors, and the independent 600 x 1800 recompute. The gap is empirical only: a legal-looking first-return profile f_3 = f_5 = 1 gives S = 0.3262 inside it, so nothing arithmetic excludes the interval and the observation is never a theorem. The open conjecture is U(a,b) <= phi^-2 for every non-shift primitive direction, which with the theorem and the Fibonacci product identity is the whole golden ceiling. Witness: gasket-ray-machine, lab/gasket-witness-weights.
• [Proved] M_n(z) <= D_n(z_1 + z_2), Conjecture W's owed first move, in one line - disjoint binary supports make m(z_1+z_2) binary below 3^n and m -> m(z_1+z_2) injective - and it is the wrong half: D_n(w) grows at rate 2, not phi, reading 4196351, 1683971, 613817, 228519 at n = 24, w = 4, 10, 28, 82 against the ceiling F(25) - 1 = 75024, so the weight enters only through the constant. Witness: gasket-ray-machine, lab/gasket-witness-weights.
• [Proved] gasket-ray-machine stated M_n(a,b) = (T^n)_{00} where its own proof gives M_n(a,b) + 1 closed paths; corrected to (T^n)_{00} - 1, and the carry bound |c| <= max(a,b) sharpened to c in [-a/2, b/2], which ties the live state count to the witness weight at floor(a/2) + floor(b/2) + 1. Witness: gasket-ray-machine.

Diagonal slice ladder

• [Conjecture] The 2-adic Smith form of M_even at base 3 and odd D has elementary-divisor valuations 0^r 1^p 2^q A with exactly one divisor >= 2^3 and sum a_i = v_2(det), so v_2 = nullity + #{a_i >= 2} + max(a_max - 2, 0) reduces the uniform bound v_2 <= n to the tent rank law nullity_2(M_even) = min_(t in T) (|D - t|/2 + 1), T = {2J(k)+1, 2J(k)+3 : k >= 2}, J(k) = (2^k - (-1)^k)/3 (exact 255/255 at odd D = 3..511, peaks J(k-1) at D = 2^k + 1, hence nullity <= ceil(n/3), troughs at the odd D with 3D nearest a power of 2, so the rank deficiency measures the 2-versus-3 carry mixing) plus a small-excess bound whose constants are domain-limited: max a_i <= 9 and #{a_i >= 2} <= 5 hold at odd D = 5..121, but max a_i <= 9 first fails at D = 127 (12 by 511), #{a_i >= 3} = 1 at D = 175 (reaches 5) and #{a_i >= 2} <= 5 at D = 183 (reaches 21); the mod-2 form is P == (1+t)^(2D-3)(1+t^3) with 1 + Dt + t^2 == 1 + t + t^2 irreducible, D = 7 is a tent trough with the whole valuation in the lone big divisor (a = {0,0,0,7}, max a_i = 7 > n = 4, a size effect), D = 5 (a = {0,0,4}) is the only other v_2 > n at odd D = 5..121, equality holds at D = 9, 15, and the fold puts the 2-content in the even block because the palindromy row c' = 0 is 2 P[D+c] entrywise and at D = 7 the odd block is 2-adically unimodular. Witness: lab/smith-cascade; slice-sign-even-half.
• [Conjecture] The base-7 certificate extends with logarithmic depth and no transient to every even D = 2..40, and the depth-death law is asymptotic rather than exact: both measured breakpoints land one even step early of D = 2 ceil(7^(K+1)/4) + 2 because the depth-K positivity frontier f_K(D) is still climbing when the window edge reaches it, so the corrected law reads K_min = min{K : (D-2)/2 <= f_K(D)}.
• [Conjecture] The central diagonal slice census of the base-3 D-dimensional Menger analog obeys a linear recurrence of order exactly ceil(D/2): the digit polynomial factors as P(t) = (1 + t^2)^(D-1)(1 + D t + t^2), the carry map c' = (c + D - s)/3 contracts to {|c| <= floor((D-1)/2)}, the symmetry v -> (2,...,2) - v gives P[s] = P[2D-s], so the Krylov subspace from e_0 sits in the reflection's +1 eigenspace of dimension floor((D-1)/2) + 1 = ceil(D/2) and the order bound holds at every D; exactness is checked at D = 2..14 by distinct eigenvalues of M_even with minimum gap above 6.9 and at D = 2..24 by nonzero Hankel determinants, and is open for general D (a square-free characteristic polynomial); controls: D = 2 gives 2^L at order 1, D = 3 gives 6, 42, 306, 2250, 16578, 122202 and A299916's 9a(n-1) - 12a(n-2) at order 2 by a route that never mentions a hexagon, D = 4 gives 6, 132, 1848, 29040, 441408, 6772128 at order 2 with dominant root (11 + sqrt(385))/2, D = 5 gives 30, 1000, 35700, 1321600, 49786200, D = 6 gives 20, 4030, 242300, 24642700, and rational Hankel elimination on nine terms reads orders 1, 2, 2, 3, 3 at D = 2..6. Witness: slice-recurrence-order; A299916.
• [Conjecture] Conjecture S, the sign law sgn(dim_slice - (d-1)) = (-1)^(D+1) at every D: the even half at bases 3 and 5 and the odd classes D != 1 mod 3 are settled on the shelf, the odd class D == 1 mod 3 beyond D = 80 is open, and the route through a named lemma, an explicit positive vector x_D with sgn((M_even x)_i - (fill/3) x_i) = (-1)^(D+1) at every index, has only the Perron vector, which supplies it numerically at every D <= 50 with worst componentwise discrepancy 9.15e-46 at 90 digits, peaked at index 0 and non-increasing, with no closed form; its two silent hypotheses, a real spectrum (exact at D <= 20, numerical at D = 2..60) and no non-Perron eigenvalue crossing fill/3 (checked at D = 2..60 against a 180-digit reference), are themselves unproved. Witness: slice-recurrence-order; slice-sign-even-half.
• [Conjecture] The second eigenvalue of the even carry block tracks the digit polynomial at -1: lambda_2 -> (-1)^(D+1) 2^(D-1)(D-2)/3 = (-1)^D P(-1)/3 with exponential convergence but never exactly (the characteristic polynomial is nonzero at that value in exact arithmetic at every D = 2..40, so lambda_2 = -9007199254740992 = -2^53 to 22 digits at D = 50, which is 2^49 * 48 / 3, is display rounding), hence rho/|lambda_2| -> (D+2)/(D-2) -> 1, measured 13/12 to 2.58e-22 at D = 50 and 1.04081632653 at D = 100, with |lambda_2|/rho = (D-2)/(D+2) to nine digits by D = 36, so the spectral gap closes and no argument may assume a fixed one; the asymptote must not be quoted at small D, where D = 4 gives a true lambda_2 = -4.310708 against -16/3, a 19% gap consistent with an O(2^(-D)) approach; measured at 420 to 650 digits. Witness: slice-recurrence-order.
• [Conjecture] The base-5 middle-digit analog (keep a cell when at most one coordinate is the middle digit 2) has P_5(t) = A(t)^(D-1)(A(t) + D t^2) with A(t) = 1 + t + t^3 + t^4, fill = 4^(D-1)(D+4) and carry rule c' = (c + 2D - s)/5, and sgn(dim_slice - (d-1)) = (-1)^(D+1) holds at D = 2..15 down to a smallest excess of 1.055e-9 at D = 15 (exact-integer sign sweep with 80-digit root refinement, agreeing with substitution-product convolution in all 16 cases at D = 2..5, L = 1..4, the D = 3 fill 112 of 125 matching the middle-digit count), yet A(-1) = 0 makes P_5(-1) = 0 for every D, so the alternating mass that carries the base-3 explanation is absent while the alternation survives, and no mechanism yet survives that. Witness: slice-sign-even-half.
• [Conjecture] The alternation is universal across base-3 designs and its phase is not: over four families at D = 2..7, Menger with at most one middle digit and P(-1) < 0 for D > 2 gives -+-+-+-, at most two middle digits with P(-1) > 0 gives --+-+-+, Cantor with no middle digit and P(-1) = P(1) gives +-+-+-+, exactly one middle digit with P(-1) < 0 gives +-+-+ from D = 3, so the phase tracks the sign of P(-1); the range stops at D = 7 and the four sign patterns rest on a prose table alone.
• [Conjecture] The excess rho_D - fill/3 decays at the rate r_inf = 1/prod_(k>=2) cos(2pi/3^k) = 1.3461220067642173 per dimension on the eigenvalue scale, 2 r_inf = 2.6922450 on the dimension scale, with a linear prefactor |delta_D| ~ A (D-1) r_inf^(-D), A -> 2/(3 prod cos) = 0.89741, from the 3-adic angle-tower product formula, matched within 1e-8 by exact rational bisection at D = 61; a third-order Richardson fit in 1/D over D >= 60 at 320 digits gave the one-step ratio 0.742874554813847413, r_inf = 1.34612251727283689 (seven true digits, the rest fit residue), even and odd extrapolations 4.5643e-8 apart and A ~ 0.897520192686, and the shape check (6A/ln 3)(D/(D+2))(2 r_inf)^(-D) = 1.47e-21 at D = 50 against the measured 1.42672e-21 tests the form and not the constant. Witness: slice-recurrence-order.
• [Conjecture] Conjecture S reduces to one separation lemma along an explicit chain: with M_even the reflection-even block of the carry automaton M[c,c'] = P[c + D - 3c'], P(t) = (1+t^2)^(D-1)(1 + D t + t^2) and fill = P(1), if every non-Perron eigenvalue of M_even has modulus below fill/3 then sgn det(fill/3 I - M_even) = sgn(fill/3 - rho), and that determinant sign is (-1)^D, exact in integer arithmetic at D = 2..20 and to D = 40, which is Conjecture S; the separation hypothesis is checked at D = 2..60 and not proved, the row-sum lemma feeding it holds for the full carry matrix on states c = 0..D and is false in the recurrent even basis (D = 3 row sums (12,4) against the formula's (8,6)), and the product formula settles the odd half without separation, so this chain is a route to the even half only. Witness: slice-recurrence-order; slice-sign-even-half.
• [Conjecture] The balanced-mask homotopy reduces Conjecture S to a one-variable determinant inequality and owes two lemmas: with a_D = (-1)^(D-1)(D-1) and Q_D = P_D - a_D t^D, the root-of-unity identity forces 1 + t + t^2 | Q_D, and fill/3 I - M_D = L_D + a_D K_D exactly with K_D = I/3 - E_D, E_D the dilation 1_(j=3i); then f_D(z) = det(L_D + z K_D) = z h_D(z) and S becomes h_D(a_D) < 0, exact at D = 2..30; det L_D = 0 is exact to D = 30 but does not follow from residue balance, because N_D lacks constant column sums in the unnormalised even basis after truncation and folding, and coefficient negativity of h_D, which settles every odd D since a_D > 0 there, is useless at even D where a_D = -(D-1) is negative, so a uniform root bound is still missing.

Spectral spacings

• [Conjecture] Every mrly fractal spectrum tested clusters rather than repels, excluding GOE and GUE, and the claim is "more clustered than Poisson", not "Poisson": two unfolding maps, both Laplacians and two independent pipelines to 4096 nodes, with random-graph and square-lattice controls separating first; a from-scratch rebuild on the Menger sponge L = 2 cell graph (400 nodes), Sierpinski L = 5 and L = 6, normalised Laplacian, a degree-12 polynomial unfolder and a 20 x 20 square-lattice control gives P(s<0.5) = 0.689 for the sponge against GOE's 1 - exp(-pi/16) = 0.17828, 0.830 for Sierpinski L = 6 and 0.594 for the square lattice; the band 0.44 to 0.57 under one unfolder does not reproduce under a third, and the sponge spectrum is 61.25% repeated eigenvalues at 1e-9, so P(s<0.5) >= 0.61 is forced by degeneracy and GOE exclusion is a corollary of degeneracy rather than an independent measurement. Witness: lab/spectral-spacings.

Hexagram provenance

• [Conjecture] No peer-reviewed source states the hexagram count: the upstream is a photograph, a video, three blog posts and one OEIS comment, and the one peer-reviewed item upstream, the Bridges proceedings paper on three-dimensional diagonal cross-sections of four-dimensional sponges, states the cut and the star-of-David holes but gives no count and no sequence (its full text searched for hexagram, A299916, 306, 2250 and 1.8184).
• [Conjecture] The hexagram slice is not scooped at any base but 3: the adjacent published work generalises the cut along dimension n and a type-k hole parameter M^n_k or draws it on a different solid, fixed to the base-three expansion throughout, and none of it treats base 5, 7 or 9; the sequel on pentagon, dodecahedron and 120-cell slices frames the Menger slice as a two-tile closed fractal family and names the directed-graph iterated function system, the same structural reading at b = 3, so the correct statement is "generalises along dimension and hole type rather than base".
• [Conjecture] The seed's anchor is self-confirming and cannot falsify anything: "regenerate the A299916 row from two independent generators" holds whether or not the slice's object is A299916's object, since two generators of the same tile census agree by construction (matrix iteration and the (9,-12) recurrence agree exactly so); the real cold check is reproducing the matrix [[6,1],[6,3]] from the published prose.

Matched random control

• [Conjecture] A self-similar slab is not a better acoustic barrier than an equally sparse random one: at levels 2 and 3 the matched random mask blocks more in every comparison, four filled-cell counts per level, 96 of 96 at 12 seeds per cell with margins of 4 to 31 sd (tree L3 fractal 0.3720 against a random mean 0.1446, sd 0.0074), every fractal value reproducing exactly (carpet L2 0.3120 / L3 0.2548, net 0.2999 / 0.2619, tree 0.3023 / 0.3720, void 0.3122 / 0.3642), so the suppression is carried by fineness and sparsity rather than by self-similarity; the repeat reused the same simulator and so tests seed-fragility rather than the physics.

Stacked hexagon moire

• [Conjecture] Layer alignment of the hexagon stack is imperfect by construction: the transparent grid fraction of the common box grows from 0.000 at n = 1 to 0.245 at n = 55, so 24.5% of the box differs between the smallest and largest silhouettes, and ray scans on the triangular lattice are aliasing-limited below band half-width ~0.01.
• [Conjecture] The exact doubling constant of the stack is 253/2160, matching the two independent branch extrapolations to 2.6e-6 and 2.2e-6 where 19/162 fails at 1.57e-4; a derivation is missing. Witness: lab/hexagon-moire. Superseded: the constant is exactly 253/2160 by the phase-map integral, see the layer-pair row under Diagonal slice stack in SETTLED.
• [Conjecture] The adjacent-pair limit is -11/135 = -0.0814815 (lab/hexagon-moire measures -0.08116 at (199, 201) and -0.08150 at (249, 251), with residue oscillation) and the gcd-echo limit is 29/135 = 0.2148148 (lab/hexagon-moire measures +0.21473 at (67, 201) and +0.21476 at (99, 297)); both are numerology-grade fits and each needs its lattice integral. Witness: lab/hexagon-moire. Superseded: -11/135 and 29/135 are exact by the phase-map integral, see the layer-pair row under Diagonal slice stack in SETTLED.
• [Conjecture] The carpet star's exact decay rate, the coefficient on (ln L)/L, is open: one frame measures -1/8 and another -0.18.

Density theorem boundary

• [Conjecture] Spanning is the wrong hypothesis for the density theorem and the sharp condition (E) is that F - F has full rank and rad(m(F)) | rad(q): six census degenerate lines satisfying it obey the formula unchanged, a prime p | m(F) not dividing q replaces the Euler factor at p by a coset-corrected one, and A(n)/k^n can fail to converge at all (base 7, F = {v : v_1+v_2 = 1 mod 3}, k = 16, index 3, period-3 subsequential limits matched to 3e-04), numerics at 1e-04 on five corrected designs plus seven null cases, provable-looking by the base-peel argument; the same failure shows at base 3, F = {0,2}^2, k = 4 > 3 with A(n) = 0 at every level against a positive predicted density. Witness: coprime-density-above-dimension-one; A396934.
• [Conjecture] The b-visible local factor of a design at a base prime is exact at every finite level: on the gasket #{x in S_n : 2 | x_1 and 2^b | x_2} = (2^(b-1)/3^b) 3^n for n >= b, by the last-digit argument, exact to the integer at b = 1, 2, 3 on every level 3..12, thirty matches with ratios exactly 1/3, 2/9 and 4/27; the Euler-product assembly it feeds is unproved for b >= 2.
• [Conjecture] Directional coprime profiles separate designs that the scalar density cannot: two base-3, D = 2, k = 5 designs with identical B(F) = 4/5 and identical predicted density 0.5471344 differ by up to 0.0327 in an eight-bin angular coprime profile at level 7 while their scalar pairwise densities differ by 0.0001001809, by exact enumeration at levels 2 to 7 with displacement multiplicities validated against N(N-1); the bin gap decays by a factor near 0.6 a level (0.375, 0.1461, 0.0946, 0.0492, 0.0327 at levels 3 to 7), so a nonzero limit is open.
• [Conjecture] The digit-restricted coprime density over F_3[t] with S = {0,1} is 9/16: measured 0.564176, 0.563471, 0.562833 at n = 10, 12, 14 by exact enumeration against (2/3)(3/4)/(8/9) = 0.5625, the correction sitting entirely at the exceptional prime t where pi(t) = 1/2 exactly against the unrestricted 1/3, the other two linear primes measuring 0.333984 and 0.333008; the finite Euler product is neither exact nor monotone, crossing 9/16 between degrees 3 and 4 and landing at 0.560193 through degree 5 while the exact no-shared-prime-below-degree-6 probability is 592189/1048576 = 0.564755, a dependence gap of -0.004563, so the density is a theorem only under three hypotheses: existence of pi_S(p), asymptotic independence over finite prime sets, and a vanishing high-degree tail. Witness: lab/function-field-density.
• [Conjecture] Mixed-radix coprime density depends on the schedule through a mod-3 effect rather than parity: at n = 12 over 8386560 ordered pairs per schedule (4096 points each), alternating base-2/base-3 gives 0.511135 with digits {0,1}/{0,1} and 0.672615 with {0,1}/{0,2}, against pure base-2 0.607874 (near 1/zeta(2) = 0.607927) and pure base-3 0.514692; both alternating schedules are exactly half even, and the 0.161480 gap comes from the divisible-by-3 fraction falling from 1/2 to 1/4, local factor 3/4 to 15/16, log advantage 0.223144, with prime 5 opposing at -0.014253 and primes 2, 7, 11, 13 identical between them; the residue mechanism for {0,1}/{0,1} is exact (every place value from position 2 on is a multiple of 6, so a mod 6 = d_0 + 2 d_1 and residues 0..5 are hit 1024, 1024, 1024, 1024, 0, 0), but no limit is established and the aperiodic staircase is untouched.
• [Conjecture] The sponge census gaps delta 20^n - A(n) measure 0.347, 0.349, 0.344 in units of 12^n, the subdominant parity-walk scale, so A(n) = delta 20^n - c 12^n + smaller. Witness: lab/sponge-visible-census.

Gasket rays and the window

• [Conjecture] Conjecture N: every primitive ray automaton of the gasket has spectral radius at most the golden ratio with equality exactly on the shift rays (3^j, 1), and the supremum off them is the supergolden ratio 1.4655713, the root of x^3 = x^2 + 1; the bound half holds for every primitive ray by the burst certificate (top edge 0.6402122 unconditionally, from 0.730424), and what stays open is strictness rho < phi off shifts and the supergolden supremum that would move the edge to 0.605303, measured on all 490 primitive rays of height <= 40 plus a fixed 766-ray sample to height 200 where every observed radius is a root of x^k = x^(k-1) + 1 or x^k = x + 1. Witness: lemma-b-pincer.
• [Conjecture] Shear class 26 carries a flat mid-octave Chebyshev residue near 1.5e-4 at n = 14, roughly 40 times its peers 98, 176 and 416, with no degenerate fibre to blame, being the class that is a graph of nothing, parametrized by the balanced-ternary integer x_2 - x_1; the excess is collinear shift-ray mass Mertens-smeared flat: code 26 builds its points as c_n minus a disjoint-support binary pair, so the golden shift family survives with M_n(3,1) = F_(n-1) exactly where its peers carry zero, and the deep excess at n = 14 is 65.7% shift rays plus 2.9% supergolden against a Mertens-predicted flat height 676 ln 3 / 3^14 = 1.55e-4 versus the recorded 1.5e-4, the carriers flat across octaves j = 7..12 including inside the proved top range, so the window-mass-in-disguise reading is dead; the "3995 points" of the first count are 3993 non-fibre plus 2 axis points.
• [Conjecture] Conjecture W and its ray twin Conjecture O: the weighted active multiplier census per octave is C 3^j, measured C ~ 120 at n = 14, unimodal in j, with activity concentrated at 3-adic depth (attainer families (3^a, 3^b +- 1), per-pair activity decaying like 0.65^K against the weight 1.5^K, so per-octave convergence is delicate); W implies Conjecture Z, hence the window (0.4475978, 1/2], and W with O closes the window entirely, both implications exact; the precursor A_(j,K) <= C 3^(j-K) fails on the deep-K families, the universal pair-prefix transfer matrix has Perron root 4, not 3, and the unweighted octave census C = 1.042, 1.136, 1.244, 1.356 at n = 13..16 is a different quantity from the weighted (3/2)^K sum W names, so W is neither supported nor damaged by it.
• [Conjecture] Conjecture Z, E(n)/3^n -> 2 exactly for the second moment E(n) = Sum_y M_n(y)^2, written E to keep it clear of the lane's ray count Z_F(n): E = T + S + R with T = 3^n - 2^(n+1) + 1 and S = 3^n - 4*2^n + 2n + 3 in closed form, so the constant 2 is exact before measurement and only R is open; R at n = 8..16 grows at about 2.54 per level, below phi^2 = 2.618, so R/phi^(2n) peaks at 3.238 at n = 12 and decays thereafter, three generators agree to n = 18, and no counterexample is known; the limit 2 has no proof and the residual's exact rate is undecided. Witness: lemma-b-pincer.
• [Proved] The shift-ray family of the gasket second moment is closed in exact form: M_n(3^j,1) = M_n(1,3^j) = prod_(r<j) F(m_r+2) - 1 with m_r = #{i in [0,n-j) : i == r mod j}, because z(3^j,1) in G_n says exactly that z is a binary string of length n-j with no two ones at distance j, which factors into j independent no-two-adjacent chains; from F(k) <= 2 phi^(k-3) for k >= 2, a two-step induction with equality at k = 3, follow M_n(3^j,1) < (3-sqrt5)^j phi^n and Sh(n) < ((4+12 sqrt5)/11) phi^(2n) < 2.803 phi^(2n) at every level, with Sh(n)/phi^(2n) -> (13+5 sqrt5)/11 = 2.198212717 by dominated convergence; the break attempt ran the family in exact Z[sqrt5] arithmetic to n = 160 and against literal enumeration of all 3^n points to n = 12, where the shift-ray share matched the closed form at every level, and an independent re-enumeration reproduced the closed form with no mismatch to n = 13 at every j and the constants to 80 digits. Witness: gasket-ray-machine.
• [Proved] The gasket ray mass laws are theorems at every level, not checks to a finite range: the live carry automata of (3,1), (1,12) and (7,3) have 2, 3 and 4 states with characteristic polynomials x^2-x-1, x^3-x^2-1 and x^4-x^3-1, so Cayley-Hamilton gives each recurrence and the first 2, 3 and 4 return counts pin it; the break attempt exhibited every reachable state by hand and by script, found the dead state -1 at (7,3) that makes reachable 5 against live 4, and confirmed the annihilator residuals vanish well beyond the automaton order. Witness: gasket-ray-machine.
• [Proved] The shelf pair automaton is not the multiplier-decomposition summand: A(s,t) counts #{z in G_n : sz, tz in G_n} while Q_n(s,t) in E(n) = T(n) + Sum Q_n(s,t) counts #{z : sz, tz in G_n}, and min(s,t) = 1 forces the two to agree, since s = 1 makes the first carry stay zero and drives every admissible digit into G; an exhaustive census at n = 9 of all 33552 ordered off-diagonal collinear pairs shows 482 of the 2656 active ordered multiplier pairs disagree, 2540 pairs (7.57%) having a witness off the gasket, the extremes (41,122), (122,41), (122,123), (123,122) with 50 witnesses each and none inside; independent re-enumeration reproduced the census from scratch. Witness: gasket-ray-machine.
• [Verified] The spectral gap survives that correction but its ceiling below 2 does not: the free-digit automaton B(s,t), reading z over all of {0,1,2}^2, has radius 3 on exactly (1,3), (1,9), (1,27), nothing in (2,3), and exactly 2 on the same twenty pairs over all 829 coprime pairs with max(s,t) <= 52, by the same exact Faddeev-LeVerrier charpoly and nonnegative-shift certificates; its largest radius strictly below 2 is 1.8488475886485 on (4,13), (4,39), (12,13), (13,36), against the A ceiling theta = 1.6956207695598, the real root of x^3 - x^2 - 2, which 44 strictly-sub-2 pairs reach or beat in the sharp split 19 strictly above theta and 25 exactly at it, the latter carrying x^3 - x^2 - 2 as a charpoly factor and the former never, with live sets reaching 167 states at both (25,52) and (31,40) against 33, so the claim that B matches A item for item is Refuted; the eigenvalue-free witness is B(4,13) having 4583352807133551 closed paths at n = 60 against 1.6956207695598^60 < 5.76e13. Witness: gasket-ray-machine.
• [Verified] The n = 9 active-pair census, previously claimed with no generator on disk, is 2656 ordered coprime multiplier pairs and 1328 unordered, 14 ordered of them the shift pairs (1,3^j) and (3^j,1) for j = 1..7; E(9) = 52212, T(9) = 18660, S(9) = 17656, R(9) = 15896, on 12170 occupied non-fibre rays carrying 18660 points, largest multiplier 2460 = floor(3^9/8), and A(s,t) return counts agree with brute force on all 1328 unordered pairs; E(n) and R(n) are regenerated for n = 1..12, filling the skipped levels R(9) = 15896 and R(11) = 124928, and an independent re-enumeration reproduced both lists. Witness: gasket-ray-machine.
• [Conjecture] The blocking lemma for the Pair Census Bound is Sum over non-shift primitive rays M_n(a,b)^2 = o(3^n): the shift rays are closed at ((13+5 sqrt5)/11) phi^(2n) and carry between 0.65 and 0.84 of R(n) at n = 6..12, so between a sixth and 0.35 of R is untouched; on the multiplier side that residue is a sum over non-shift active pairs numbering 10, 30, 106, 332, 1010, 2642, 7564, 20934, 57858 at n = 4..12, growth about 2.77 a level with largest multiplier exactly floor(3^n/8), so lambda <= 2 per pair buys nothing without a constant C(s,t) summable against that count. That per-pair summability is now Proved dead by construction and the door is restated in the witness coordinate as Conjecture W sharp, R(n) = O(phi^(2n)), with the weight-four orbit and the shift family both closed and only summability over the witness weight owed. Witness: gasket-ray-machine.
• [Conjecture] Statement (A) is the exact averaged theorem the ray machine needs: for primitive non-shift (a,b) with 3^j <= max(a,b) < 3^(j+1), sum M_n(a,b) <= C 3^(2j) lambda^(n-j) poly(n) with lambda < phi inserts into the octave sum and gives beta > 1/(2 - log_3 lambda); it is weaker than a uniform non-shift spectral gap and far stronger than any average of rho, and its tail form requires the octave-j count of rays with rho >= t to be at most 3^(2j - I(t)j + o(j)) followed by an optimization over t; two routes are named, a large sieve on a bounded local deficit observable Fourier-expanded over the ray's residue modulus and a finite-state fractional-moment operator, with three obstructions to the sieve (varying state spaces with no common separated frequency family, the Cauchy-Schwarz loss of the square root of the ray count in passing from an L^2 average to L^1 octave mass, and an unweighted octave count the sieve's measure must match), and Turan power sums are ruled out, since they lower-bound maxima where an upper bound for a positive sum over many nonnegative matrices of varying dimension is needed. Witness: lemma-b-pincer.
• [Conjecture] The weighted active multiplier census W_j(n) <= C 3^j is not numerically stable across the two known levels: at n = 9 the exhaustive active-pair census gives A_j/3^j = 4.000, 6.667, 6.370, 4.025, 1.794, 1.141, 0.368 and W_j/3^j = 7.500, 17.750, 23.719, 25.041, 12.841, 12.097, 5.837 for j = 1..7, a peak of 25.0, while the n = 14 summary reports a peak near 113, so one level supports a constant and the two together do not, and the n = 14 tally is not in this tree; this weakens but does not refute Conjecture W, since W and O together closing the window is exact and W and O themselves remain untested.
• [Conjecture] The 3-power family of the second moment sums in closed form: S(n) = 2 Sum_(j=1..n-1) Q_n(1, 3^j) = 3^n - 4*2^n + 2n + 3 from Q_n(1, 3^j) = 3^(n-j) - 2^(n-j+1) + 1, so E(n) = T(n) + S(n) + R(n) with T + S = 2*3^n - 6*2^n + 2n + 4 exactly and the whole of Conjecture Z's constant 2 is accounted for before any residual is measured, leaving one statement about R; the identity is machine-checked at n = 3..17 and R = Z - T - S re-subtracted against the E list at n = 13, 14, 16 agrees.
• [Conjecture] The Pair Census Bound R(n) = o(3^n) over ordered collinear non-fibre gasket pairs whose multiplier ratio is not a power of 3 is the only unproved step to E(n) = O(3^n), hence to Conjecture Z and the window (0.4475978, 1/2]; the dominant carrier is the shift-ray family pairing with itself, now closed at ((13 + 5 sqrt5)/11) phi^(2n) and carrying only 0.65 to 0.84 of R, every other ray having spectral radius below phi, and any proof must use the multiplier-specific automata, since the universal pair-prefix transfer matrix has Perron root 4 and not 3. Witness: lemma-b-pincer.
• [Conjecture] Per-octave occupied ray counts by exact exhaustive enumeration at n = 14, 15, 16 total 3,151,658, 9,491,966 and 28,545,342, with occ(j,n)/3^j peaking at j = 8 in all three at 2.217, 2.492 and 2.740 and low octaves j <= 4 identical at all three levels; the table cannot be joined to the n = 13 per-octave table under the half-open convention 3^(j-1) <= max(a,b) < 3^j, the discrepancy not being a uniform label shift (4 rays with max = 3 placed at octave 1 where the interval forces octave 2; 336 against a recomputed 294 in the next bin), and the fibre-ray convention (1,044,842 with the two axes, 1,044,840 without) is not part of this mismatch.

Moment ladder and Lemma B

• [Conjecture] The moment ladder's rows above the tenth approach the wall and stop: beta_0^(2K) reads 0.447838092, 0.447904613, 0.447923402, 0.447928788, 0.447930346 at 2K = 12, 14, 16, 18, 20 from Perron roots 59307.487289, 532101.317617, 4784678.13057, 43051182.4466, 387432198.159, the twentieth row lying only 6.42e-7 below the universal wall 2/(3 + log_3 5) = 0.447930987882; these are floating eigenvalues of exact integer matrices, not interval-certified, and the wall above them is separately settled and unaffected. Witness: lab/dimension-one-ladder.

Fill polynomials

• [Conjecture] The discriminant staircase is gapless and counts 2(D-1) new discriminants per dimension: the negative fundamental discriminants (d = 0 or 1 mod 4) carried by the irreducible quadratic factors of fill polynomials at dimension D form a gapless initial segment of length D(D-1), D = 2 giving -3, -4, D = 3 adding -7, -8, -11, -12 for 6, D = 4 adding -15, -16, -19, -20, -23, -24 for 12, with 20 at D = 5, exhaustive over 17424 signatures, gapless to -40; at D = 6, exhaustive over 1053696 signatures, the peeled-remainder reading is gapless from -3 to -63 with length 31 and the reading over every irreducible quadratic factor gives length 80 to -160, deepest -899, so the D(D-1) count law predicting 30 is Refuted at D = 6 under both readings while the run stays gapless; gapless and unbounded forces every imaginary quadratic order to appear at some finite D, the geometric content of the imaginary completeness conjecture; exhaustive at D = 2..6 by exact factorization. Witness: lab/fill-polynomials.
• [Conjecture] The complement on the parity cube swaps the two arithmetic tribes: for a D-dimensional design with fill P(n) and void V(n) = (2n+1)^D - P(n), when one factors completely over Q (divisor tribe, zeta, d-counts) the other carries an irreducible factor (ball tribe, theta, representation numbers), so the integer census of a design is two sequences, one per tribe; instances: carpet (1,2,0) has fill (3n+1)(n+1) and void n^2, (0,2,0) has void 2n^2+2n+1 (the Gaussian norm, disc -4), (0,1,0) has void 3n^2+3n+1 (the Eisenstein norm, disc -3), the sponge (1,3,0,0) has fill d(240^n) and void core the linear 4n+3, (1,0,0,0) has fill (n+1)^3 = d(30^n) and void core 7n^2+9n+3 (disc -3), (1,2,2,0) carries disc -16 on the fill and -3 on the void, both tribes at once, and (1,2,1,0) is self-dual with both sides divisor; P + V = (2n+1)^D and V(0) = 0 forcing V = n Q(n) are exact, generic irreducibility of Q is a sketch, and the self-dual (1,2,1,0) is a known exception class. Witness: lab/fill-polynomials; divisor-avatars.

Half-ball chords

• [Conjecture] The mismatch theorem: a design's coprime density equals a half-ball flat-face probability at D = d = 2 and nowhere else, since design densities are rational multiples of 1/zeta(D) while Version L gives rational/Pi^2 at even d and a pure rational at odd d and Version H gives Q + Q/Pi^2 at even d and Q + Q Pi at odd d, so even D >= 4 is blocked by Lindemann and D = 3 against Version L by Apery, both unconditionally; D = 3 against Version H is conditional on zeta(3) not being algebraic over Q(Pi), odd D >= 5 on zeta(D) irrational (Rivoal and Zudilin give it only for infinitely many odd D), and the D = 2 uniqueness half is numerical, 11 base-2 and 502 base-3 designs against every Version L value to d = 24, base-2 numerators 4, 16/3, 6, 8, exactly one match, 16/3 at d = 2 carried by 3 designs. Witness: lab/half-ball-mismatch; coprime-density-above-dimension-one; A395134.
• [Conjecture] The Version H half-ball family, d uniform points and the hyperplane through them, has exact values 4 - 19845 Pi/16384 at d = 3, 4 - 549978112/(14189175 Pi^2) at d = 4 and 16 - 178919214166875 Pi/35184372088832 at d = 5, so odd d carries Pi^1 where Version L carries a pure rational, by an unoriented-normal Blaschke-Petkantschin reduction integrated in closed form, quadrature at 60 against 80 digits agreeing to 2.3e-62, 7.2e-64 and 1.5e-63, and an independent 10^8-sample random-point estimate whose deviations 2.39e-6, -6.74e-6, -3.55e-6 sit inside one sigma of 3.96e-5, 2.6e-5, 1.54e-5; d = 6 and d = 7 are exact too, 16 - 10363195833496113250304/(65656392092180764875 Pi^2) = 0.0074784083 and 64 - 403492347953923610203877211975 Pi/19807040628566084398385987584 = 0.0021206659, so the parity law "even d gives Q + Q/Pi^2, odd d gives Q + Q Pi" rests on six terms and no proof. Witness: lab/half-ball-mismatch.
• [Conjecture] No Euclidean body reproduces the bracket: a sweep of composite bases finds B(F) taking 1/2, 5/8, 3/4, 7/8, 1 across twenty-five base-4 designs of identical dimension 1.5, so B(F) is where a design's geometry lives, but no body whose chord-power integral reproduces B(F) was found, and the mismatch theorem makes the search futile above D = 2; a failed search, not a proof of nonexistence, and the base-4 sweep has no generator in lab/. Witness: lab/half-ball-mismatch for the futility only.

Robin corridor

• [Conjecture] The Robin corridor is orthogonal to Robin's difficulty, not merely hard: a design of dimension D is an integer of exactly D prime factors and level raises the exponents, so the tree varies exponents at fixed prime support, while all of Robin's difficulty lives at growing support omega(n) -> infinity; fixed support is the classical easy half, settled by prod p/(p-1) bounded against a divergent log log, an obstruction read off the definitions. Witness: divisor-avatars.

Complex dimensions

• [Conjecture] Every one-base design is lattice, so the Lapidus-Maier machinery is not out of reach but empty: zeta_L(s) = 1/(1 - k q^(-s)) puts the complex dimensions on one vertical line of period 2 pi/ln(q), and (ISP)_D, the Riemann hypothesis in the language of fractal strings, has no content on the degenerate case, several incommensurable ratios being the change that gives it content; the lattice half is checked on this tree's own poles and folding tables, the (ISP)_D half is a literature reading not yet checked at source, and the dichotomy is a theorem for strings only, so dimension two and above is open outside the pluriphase class, inside which the carpet and the interior-hole designs are settled. Witness: lab/complex-dimensions.

Acoustic barrier

• [Conjecture] The level trend of the self-similar acoustic barrier is fineness, not self-similarity: a random tile at matched solid fraction and cell size, three seeds, r = 4, also falls with level and blocks more at every L2 and L3 - random against self-similar at L2 / L3: carpet 0.264 / 0.074 against 0.390 / 0.128, net 0.229 / 0.099 against 0.467 / 0.215, tree 0.221 / 0.066 against 0.781 / 0.666, void 0.211 / 0.104 against 0.270 / 0.235 - with a seed spread of 0.006 to 0.037.

The diagonal cut

• [Conjecture] Cook's (2011) Menger-slice code cuts along the normal (1, 1, 0.5), not the centroid diagonal, so the base-3 diagonal cut is not Cook's slice; the reference row scoring the cut as Cook's is kept at yes with that correction recorded beside it, and the row once scored yes in error now reads no.

Transparent fraction

• [Conjecture] The common-box transparent fraction does not grow to 24.5%: GRID(n) = (n-1)/(4n) for odd n reproduces 1/6, 2/9, 13/54, 27/110 at n = 3, 9, 27, 55, reads 0.2495463 at n = 551 and 0.2499504 at n = 5041, and its limit is 1/4; 24.5% is the n = 55 value, and at n = 0 the normalization is undefined.

Carpet star

• [Conjecture] The carpet-star decay coefficient is not exactly -1/8: it depends on the registration and band-normalization frame, the family frame through L = 400 giving slope -0.1242 and the ideal frame through the same L = 400 giving -0.18 to -0.19 and still drifting; only the (log L)/L decay order survives the frame change.

The Farey stack

• [Conjecture] The Baez-Duarte coefficients c_k = sum_n mu(n) n^{-2} (1 - n^{-2})^k = sum_{j=0}^k (-1)^j C(k,j)/zeta(2j+2) from an independent Mobius sieve to n = 10,000 read -0.316011506 at k = 1 to -1.68003e-5 at k = 1000 against a 450-digit reference -1.65958e-5, difference 2.04521e-7; the sieve agrees with the test vector [1,-1,-1,0,-1,1,-1,0,0,1] and with a second linear sieve at every integer through 10,000, counts 3053 minus-ones, 3917 zeros, 3030 plus-ones; consistent with the criterion and evidence for the Riemann hypothesis of exactly nothing.
• [Conjecture] S_N = sum_{k=1}^N (-1)^k C(N,k)/zeta(2k) is not a Riemann-hypothesis criterion tending to zero: it has the wrong zeta shift, omits the j = 0 term and tends to 2 - S_100 = 1.843329, S_500 = 1.967518, S_1000 = 1.983699; the sequential Baez-Duarte coefficient is c_k = sum_{j=0}^k (-1)^j C(k,j)/zeta(2j+2), the Nyman-Beurling distance d_N = inf ||1 - D_N||^2 is not the coefficient sequence and needs its own basis and Gram matrix, sum_{j | k} mu(j) is the Mobius inversion identity (1 at k = 1, 0 after), and direct binomial evaluation at 80 digits is nonsense by k = 500, so 450-digit arithmetic or the Mobius series is required.
• [Conjecture] Under the convention Q = 3^L the Landau discrepancy reads 0.166667, 0.549206, 1.150760, 2.118500, 3.187070 at Q = 3, 9, 27, 81, 243, computed in exact rationals by two routes that agree; D_Q/sqrt(Q) stays in [0.0962, 0.2354] and the last-three log-log slope is 0.464, consistent with O(Q^{1/2+eps}) and discriminating nothing, since five nested deterministic points cannot test a statement quantified over every positive epsilon.

Integer census and avatars

• [Conjecture] No geometric observable beyond dimension tracks Robin's inequality along the colossally abundant numbers: over the first 50 the Robin ratio is governed by D alone (Spearman rho = 0.9906 on indices 13-50, adjusted p = 2.93e-32) while the fill polynomial adds nothing (normalized-fill coefficient 0.0190, p = 0.649, AIC worsening from -116.34 to -114.56), because every exponent equal to 1 contributes zero to (a_i - 1), so a new largest prime raises D and doubles k while fixing every nonzero coefficient, the maximum nonzero degree being 6 while D reaches 34; 38 dependent points over 6 <= D <= 34 are a corridor, the fit 1 - R = 0.03611 exp(-0.03089 D) at R2 = 0.960 is descriptive only, the ratio is not monotone (minimum 0.964531 at index 13, n = 21621600, 5 of 37 later transitions non-increasing), and nothing here bears on the Riemann hypothesis.
• [Conjecture] 10 is the smallest positive integer that never occurs as a base-2 design fill count at side number 2, the fill counts through 1000 being 1, 2, 3, 4, 5, 6, 7, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 125, 128, 216, 243, 256, 343, 512, 625, 729; fill 10 does occur at side 4 for a base-3 2D design; the catalog behind the original count had no generator.

The slice ladder: rate and dead routes

• [Conjecture] The roots-of-unity circulant correction Q_D = fill/3 + (2/3)(-1)^(D-1)(D-1) cos(2 pi D / 3) is worse than bare fill/3: mean absolute error 11.05299392007777488084818 against 0.1511524922667271359757626 over D = 2..50, a factor 73.1248, with |rho - Q_D| / |rho - fill/3| reaching 1067922.7 at D = 50; its sign matches (-1)^(D+1) only when D = 0 mod 3, 16 of 49 cases; the order-D root-of-unity term is cancelled by finite-boundary effects, the empirical correction factor collapsing to -9.363982332e-7 at D = 50, so any model of the excess must derive the boundary cancellation.
• [Conjecture] The three-block DFT decomposition of the carry matrix does not exist: the span of 1, omega^c, omega^(2c) is not invariant for any D >= 5 (relative Frobenius residual 0.318 to 0.523), carry residues mod 3 are coupled for every D >= 3 so M does not commute with diag(omega^c), and the three-root average (P(1) + P(omega) + P(omega^2))/3 misses the Perron root by -9.191 to +11.263 while the true excess is -0.0435 at D = 20; the mod-3 block version has off-diagonal Frobenius mass of order one, ratio 0.609 to 4.111 over D = 3..30 with slope -0.00162 +- 0.00773 per D, p = 0.836, and a Schur correction at fill/3 positive for every D, 0.343 of fill/3 at D = 30; one exact row-sum identity survives.
• [Conjecture] The saddle-point route is closed: the transfer operator is coefficient decimation, (Mv)(c) = [t^(c+D)] P_D(t) V(t^3) on a finite carry window, not multiplication by a scalar symbol; geometric vectors z^c are not eigenvectors, the all-ones vector is the only reflection-even one and is not an eigenvector either; on the unit circle max|P_D(e^(i theta))| = P_D(1) = fill, whose cube root is exponentially smaller than rho_D, while max|P_D|/3 is exactly fill/3 and misses the whole effect; no non-tautological f_D(theta) with rho_D = max|f_D| was found.
• [Conjecture] Induction on D is closed from both ends: the same-size correction between M_D and M_(D+1) at odd D has full rank at every D = 3..19, determinants from -54 at D = 3 to -441065669103434214513656226772598887664331096 at D = 19, so the matrix determinant lemma has no low-rank update to consume; the threshold determinant sequence d_D satisfies no recurrence surviving holdout - constant-coefficient orders 1 to 6 with degrees 0 to 5 on the full sequence and each parity subsequence, all 62 identifiable holonomic pairs with r s <= 40, normalisation by fill^n and by D^beta for beta = -4..4, Berlekamp-Massey over three primes at maximal linear complexity (20 for 39 terms, 10 per parity), the one determined fit (order 4, degree 2, odd subsequence) failing at D = 35, 37, 39; the 2-adic valuation of d_D fits none of the tested elementary forms.
• [Conjecture] Cauchy interlacing is closed: over all 27 pairs 2 <= D <= 28, M_even(D) is not the upper-left block of M_even(D+2) and none of the (n+1)^2 row-column deletions of the larger matrix is the smaller, so no bordering u, v, alpha exist; the eigenvalue chain holds for every even start and fails for every odd one, witness lambda_1(3) = 7.372281323269 against lambda_2(5) = 16.965208741322; the threshold count it was meant to prove is nevertheless exact on D = 2..30 - no eigenvalue above fill/3 at even D, exactly one at odd D.

b-visibility

• [Conjecture] The gasket's b-visible density is (1 - 2^(b-1)/3^b) Prod_{p odd}(1 - p^(-(b+1))), not (8/9)/zeta(b+1) for every b >= 1: the local factor at 2 is 2^(b-1)/3^b, exact to the integer at b = 1, 2, 3 on every level 3..12, and the ratio to 1/zeta(b+1) is (1 - 2^(b-1)/3^b)/(1 - 2^(-(b+1))), which is 8/9 at b = 1 ((2/3)/(3/4)) and at b = 2 ((7/9)/(7/8)) only - 368/405 = 0.9086420 at b = 3, 2336/2511 = 0.9303067 at b = 4, climbing to 1; the two formulas part at b = 3, 0.8395292 against 0.8212786, and exact enumeration reads 0.8427119 at level 12, falling about 0.0022 a level toward the former; b = 1 is the proved 16/(3 Pi^2).

Coprimality at dimension one

• [Conjecture] No fixed modulus decides mixed-radix coprimality: the smallest moduli labelling coprimality exactly on the n = 12 sets are 27994 and 20736, at which all 4096 values occupy distinct residues, an encoding of the finite set rather than a transfer matrix; a prime not dividing M is invisible modulo M, so no fixed finite state space decides coprimality on an unbounded family, and a finite matrix tracks a fixed finite prime set exactly and nothing beyond, which is why the truncated Euler product through 13 misses by 0.008977 and -0.031676 on the two alternating schedules.
• [Verified] Occ_T at the cyclotomic T is exactly the set {(3^t + 1) a_S} with its complements, of size 2(2^(t-1) - 1) whenever R_k is prime and 2, 6, 12, 30, 62, 100, 254, 510 at t = 2..9, and max_T |Occ_T| m_T / 2^k reads 4.562 to 376843.283 at odd k = 5..19 at that T every time; an unconditional statement needs #{S in [1, t-1] : gcd(a_S, R_k) = 1} >= 2^t / poly(t), nowhere proved. Witness: lab/ratio-set-saving, ratio.py lifts --kmax 19 21 s.
• [Verified] The lift union to k = 19 satisfies U_k <= Sum_T |Occ_T| = agg_k L_k Phi_k with L_k = Sum_T 1/m_T < 3/2 Proved; over k = 11..19 agg_k sits inside [1.01748, 1.11457] with no trend, L_k inside [1.41043, 1.41724] and U_k / Phi_k rises monotonically across [1.19611, 1.36517], so U_k = O(2^k) is the boundedness of agg_k alone. Witness: lab/ratio-set-saving, ratio.py lifts --kmax 19 --zmax 15 10 min 43 s.
• [Verified] The absolute-value route on the u != 0 Fourier terms of the lift count is dead: Sum_T (1/m_T) Sum_(u != 0) |F_T(u)| / 2^k reads 1.3839 to 7.9155 at k = 5..11, step ratios all above 1.26. Witness: lab/ratio-set-saving, ratio.py agg, the Abs column.
• [Verified] The cut-free aggregate agg'_k = Sum_T (N_T - 2)/(2^k L_k) sits inside [1.03919, 1.3403] and M_k / 2^k inside [2.47278, 2.89356] over k = 11..19, no upward trend. Witness: lab/ratio-set-saving, ratio.py agg, 20 s.
• [Verified] Writing b(z) for the number of k-blocks the minimal witness lift m(z) R_k fills, U_k = #{b(z) <= 2} (Proved) and the depth-3 census V_k = #{b(z) <= 3} gives (U_k, V_k, Z(R_k)) = (2342, 2350, 2360), (1618, 1624, 1634), (10280, 10310, 10388), (10278, 10310, 10440), (35566, 35630, 36190) at k = 11..15, so depth 3 captures 8, 6, 30, 32, 64 of the deep tail 18, 16, 108, 162, 624, a share falling 0.4444, 0.375, 0.2777, 0.1975, 0.1025, and the one-position lifts add nothing at any k <= 13. Witness: lab/ratio-set-saving, ratio.py tail.

Conjecture S: the even-half transient

• [Conjecture] Every weighted-L2 certificate for the even half fails, the transfer norm being at least sqrt(3) fill for every positive weight, and two Abel pairings fail with it.

Conjecture S: even half

• [Conjecture] The row certificate's sign law at base 5 is periodic, not one-sided: (v^T M^k)_j >= 0 iff dist(j, 5^(k+1) Z) <= (5^(k+1)-1)/4 (witness D = 40, k = 1, j = 19 positive), the threshold being the carry-drift radius around every multiple of 5^(k+1), not just around 0. Witness: slice-sign-even-half.
• [Conjecture] Three negatives on the base-5 even half: quintupling resummation is structurally empty, the two orbit relations of the base-5 tower summed over complete residue systems returning exactly the one-step identity 5 b(L+1) = fill b(L) - (D-1) V(L), the nontrivial comb mapping into the trivial comb whose orbit factor is 1 and the correctly normalised scaling limit of V being 0 = 0; any envelope bounding numerator and denominator independently dies at psi = 0, the cone having zero width at both census points 2 pi/5 and 4 pi/5 (both slack summands nonnegative with nonpositive sum), so only curvature-coupled envelopes remain; and the neutral Gaussian width of the transfer at psi = 0 is exactly a* = m2/(24 fill) = Var(digit sum)/24 = 5D(D+3)/(48(D+4)), not 5D/48, which explains the measured upward drift of sig2/D toward 5/48.
• [Conjecture] At every odd base q >= 5 the even half falls to the base-5 template at depth O(log D) with no transient: V-towers at q = 5, 7, 9, 11, 13, even D = 8, 12, L <= 25 show no dip anywhere off base 3. Witness: slice-sign-even-half.

The 2-adic Smith cascade

• [Conjecture] Layer 2 is read by the Bockstein pairing B(z,w) = (1/2) z^T M w-hat mod 2 with L_2 = nullity - rank(B) and the closed coefficient form (1/2)[x^(6R+1)](P What Zetahat) (49/49 at odd D = 5..101, the coefficient identity exact at about 1.3k pairs, a corollary of the extraction form), and the layer flag V_k = red_2(ker(M mod 2^k)) is a contiguous step-3 degree run for all k <= 9 at D <= 201 (99/99), not always top-anchored (witness D = 29: mod-4 corrections pinned at the window top break shift-closure).
• [Conjecture] Smith(core) = Smith(even) ∪ Smith(odd) as multisets at base 3 (D = 5..91); it fails at base 5, D = 31: even {1,3,4} + odd {1,2,2} against full {1,1,1,2,3,5}.
• [Conjecture] The base-5 nullity has no bounded tent: the mod-2 nullity valley floors rise linearly, 1, 2, 2, 4, 8, 14 at D ~ 19 * 2^k, about D/38, with peaks about 0.1 D, so the Jacobsthal tent with floor 1 is a base-3 phenomenon; at large odd class-D the profile is rigidly [1] + [2]^(L_1-2) plus two tail terms, almost all 2-torsion in one layer, a parity split with no mechanism.
• [Conjecture] Off a maximiser the profile is two-tier, [(j-1)^p, j^q, spike] with p(i) = 2i - 1 marching in from the site and plateau length 2 in the site's own L_j (5 sites, 23 rows; D = 1379..1383 mirrors D = 689..693), D = 1449 (L_2 = 41, tail [2^39, 4, 7]) is an ordinary j = 2 flank row with a one-unit tier-height excess at q = 1, and spikeless rows exist - D = 1373 is a pure flat block, tail [4,4,4], L_5 = 0, double-sourced at a different precision and Smith-free.
• [Conjecture] The cascade recursion: the layer-3 law is the layer-2 window law one level down - in the quotient coordinate u = c/g_2 (the layer-2 window is deg u in [0, L_2 - 1]), V_3 is again a divisor-plus-ceiling window with tent parameter two Jacobsthal indices down, c_3 = 2J(e-4) - 1, delta = C_2 - C_3, P = C_3 - deg g_2, j = deg g_3 - deg g_2 = max(0, 2P - c_3), L_3 = min(P + 1, c_3 + 1 - P), tower ladder (delta, c_3) = (0, 2J(e-4)-1) while L_2 <= 2J(e-4) then (2J(e-4), 2J(e-5)-1); 40/40 on rows with L_3 >= 2 (odd D = 175..401 complete plus 701..707, 735..741, 363, 365, with window, dimension and nesting exact and no contiguity break), six rows predicted before computation (D = 735..741 with a never-seen delta = 22, seam rows 363 with j = 9 and 365); j is always 0 or odd, the layer-j tent height is J(e - 2(j-1)), which on the k = 1 slot is J(k_oct + 2 - 2j) - the amplitude law derived for j <= 3 - sites tie to the octave troughs (K = (D - t_k)/2 at 38/38), and c_4 = 2J(e-6) - 1 puts the first L_4 >= 2 at exactly 689; g_3/g_2 is not always a monomial nor Fibonacci-shaped (D = 481: c_2(z^2); D = 497: (1+z)^2); further, t > 0 => L_3 = 1 on all 22 rows of the t > 0 region of octave b = 10; unswept: 403..471, 517..699, 709..733, 743+, and the layer-4 window at 689..693. Witness: slice-sign-even-half.

Spin

• [Conjecture] The near-degenerate corner ripples beyond the segment case: de-duplicated to transpose classes, the equal-fill class pairs sitting inside their own drift bar number 13 at level 6 and 6 at level 7, at gap-to-bar ratios 0.71 to 0.95, the tightest 287 against 315 at fill 6, gap 0.03574 on bar 0.04543, two designs that differ by moving one cell from (0,2) to (1,2); six survive both levels, 287-315, 63-123, 123-187, 31-59, 437-485, 37-261, and the count moves with the level and the estimator, so the list is a phenomenon and not a census. The ripple's Fourier coefficient at frequency 2 pi / log 3 should be a linear functional of the digit set whose kernel is what collides. Witness: lab/spin-census.
• [Conjecture] The powder falls as k^-D - at level 7 with pad 4096 the slopes -1.37986, -1.51762, -1.73012, -1.83071, -1.97886, -2.00433 sit within 0.24 of -D over both pads, but the agreement is inside the instrument's own spread: sliding a three-period window a quarter period at a time moves the slope by 0.16 to 0.45, and doubling the pad to 8192 moves 127 from -1.73012 to -1.81607, 255 from -1.97886 to -2.03225 and the carpet from -2.00433 to -2.12289, with no monotone approach to -D. The log-periodic ripple is not resolved either, the folded residual swinging 1.5 to 4.4 in ln power because the ring average of a lattice point set is spiked on the norms of A001481. Witness: lab/spin-census.
• [Conjecture] The axes are the sponge's only deficient shadow directions; the window checked is |v| <= 3. Witness: lab/spin-census.
• [Conjecture] new(n) sqrt(ln n) / n^2 converges to sqrt 2 K prod_(p | n) (1 - 1/p^2) along each radical class, K the Landau-Ramanujan constant; the Mobius sum is proved but B has no closed form, so the Gaussian Farey carries a transcendental constant where the Farey carries none. Witness: lab/spin-census, A064533.

The registry's integers

• [Conjecture] That a missed integer is written by no row at any depth. The miss set is a statement about the rendered window: 6802 rows are cut by the 48-term cap, the 96-term lower bound already moves at least 765 misses across, 269 among them, and the cost of a true deeper census is cubic in the cap on the dimension-2 side grid. Witness: lab/integer-census.
• [Conjecture] That the miss set is new to the OEIS. It is clean only in its dense head: no record of the dump carries a 4-term window of the miss set at offsets 0..416, the first hit being offset 417 in A049537, and above that the miss set hits near-interval records - 852 hits at k = 4, 130 at k = 10, 37 at k = 15 and 15 at k = 20, the 20-term witnesses A112820 and A118471. The absence rests on a dump, which is a snapshot, so it stays a conjecture under the standing caveat. Witness: lab/integer-census, A049537, A112820, A118471.
• [Conjecture] That the write-once set is absent from the OEIS. The 2897 integers written by exactly one row have no hit at any offset of any record at k = 4, 10, 15, 20, a cleaner absence than the miss set's because the once set is thin where the miss set is an interval complement; the same dump caveat applies. Witness: lab/integer-census.
• [Conjecture] That the miss set has no arithmetic characterisation. No modulus to 64 separates written from missed, the written share has no common growth order, and the set is closed under nothing; the finiteness bound is the only theorem the lane offers. Witness: lab/integer-census.
• [Conjecture] That no bounded union of named families reaches the written tail. Above 30000 the rows' written sets collapse to 953 distinct families of which 875 own a tail integer no other family writes, covering 2005 of the 2174 tail integers, so every cover needs at least 875 families; the families are de-duplicated by written set and not by generating rule, so 953 is itself a lower bound on the number of rules. Witness: lab/integer-census.
• [Conjecture] That no reparametrisation of the multiplicity function a(n) = rows writing n is submittable. It is absent from the dump at every offset tested, but it is a reading of the registry's own shape - the tier mix, the cap and the ceiling - rather than a function of n, so its terms move with the instrument. Witness: lab/integer-census.

The component exponent

• [Conjecture] Which words with no letter frequencies carry a component exponent at all, now that all 105 letter pairs have closed forms and interior frequency settles the rate: over (3, 7) the tripling word W_(k+1) = W_k 7^|W_k| 3^|W_k| keeps both letters at lower density 1/4 and still has its prefix rate range over [0.4792, 1.4379] in log 2 units on 1024 <= L <= 4096 with no narrowing, so positive lower density is the wrong hypothesis, but that is measurement and not a proof that the limit fails. Witness: lab/magic-words, connectivity.md.
• [Conjecture] The accumulation set of the saturation comp/fill along Thue-Morse over a gasket-against-domino pair, plausibly the attractor of the two affine chart maps read along the word; the sampled value 0.2325367033 at L = 4096 is a term of an oscillation and is neither a limit nor a maximum, the exact maxima at L >= 5 being 43397/186624 and 151/648 under the two readings. Witness: lab/magic-words, connectivity.md.
• [Conjecture] Whether any alphabet of three or more letters, or any other order-sensitive observable, makes the component-style exponent depend on more than the letter frequencies, which is what a non-stationary result would need; at interior frequency on two letters it provably does not, on any of the 105 pairs. Witness: magic.md, connectivity.md.
• [Conjecture] Whether the same suffix recursion that closes the 46 reproduces the 59 forms proved by other means, which would collapse the whole table to one lemma, and whether the Euler, boundary and holes series of Hankel ranks 4, 8 and 11 answer to it as well. Witness: connectivity.md.

The tile monoid

• [Conjecture] The cut set L(C) is closed under gcd, with zero failures over every non-empty subset of a line at N = 1..20 and over all 339795 side-12 plane-code composites and no proof; it is the one missing structural fact, since with it L(C) is a meet-subsemilattice of the divisor lattice and the canonical name closes, and without it there is no counting theorem at non-prime-power side, where 171 and 15159 are enumeration rather than formula. Witness: lab/code-factorisation, magic.md.
• [Conjecture] Two tiles commute under the Kronecker product exactly when they are powers of one common tile or the members at their two sides of one scale-free family, the one-cell case being settled by a(n - 1) = b(m - 1) and the general case tested only at (2,3) in two dimensions and at ten side pairs in one; relatedly, whether the diagonal and the antidiagonal are the only permutation tiles factoring in both radix orders at every coprime split. The next coprime test needs all 2^25 side-5 codes, so this has to be settled by proof and not by search. Witness: lab/code-factorisation, magic.md.
• [Conjecture] The three-family description of cross-shape collisions at a coprime shape - axis-separable rectangles, tiles with a fill-1 outer factor, and the diagonal pair - is exhaustive at side 6 and untested anywhere else. Witness: lab/code-factorisation, magic.md.
• [Conjecture] An intrinsic description of which tiles are Kronecker products, rather than the block test's algorithm and the published decomposition graph; and what the irreducible letters of composite side do, now that they are known to exist and to be generic, which is the question the plane-code word census could not see. Witness: lab/code-factorisation, magic.md.
• [Conjecture] The diagonal embedding D -> {(x,x) : x in D} is an injective, divisor-closed embedding of the one-dimensional digit-set monoid into the tile monoid preserving cut sets and irreducibility, and a diagonal tile is axis-separable only at one cell; so a line sweep is a tile sweep, the whole non-uniqueness phenomenon already lives in one dimension, and it lifts to tiles no rectangle can explain. Witness: lab/code-factorisation.

Crop census

• [Conjecture] The curved-slice dimension: the log_3 cut-ratio exponents of the inscribed circle on the carpet read 1.140, 0.909, 0.899, 0.951 and of the sphere on the sponge 2.166, 1.579, 1.705, hovering near the straight-slice yardsticks d - 1 = 0.8928 and 1.7268 - yardsticks by analogy only, since Shmerkin 2019 and Wu 2019 cover intersections of xp- and xq-invariant line sets with p, q multiplicatively independent, not same-base carpet slices, straight or curved; five levels decide nothing. Witness: lab/crop-counts, crop.md.
• [Verified] The pointwise factor widens with decelerating drift: carpet maxima rise 1.125000 to 1.518945 by increments 0.140625 down to 0.000808 over eight windows r = 1..6560, minima in [0.588115, 0.900000]; sponge maxima 1.350000 to 1.673315 over five windows r = 1..242. Witness: lab/circle-crop factor lines.
• [Verified] The window multiplicity kappa = W(R)/((m-1) M(R)) brackets to [1.247746, 1.248322] on the carpet at r = 2187..6560 and to [1.417534, 1.445977] on the sponge at r = 81..242, still climbing there; the [1, 2] bound is asymptotic, the exact slack being (C(3R) + C(R))/((m-1) M(R)). Witness: lab/circle-crop mean lines, form_low = -4.744629 at r = 1..2.
• [Verified] The l^1 mass Lambda_L reads 1.000000, 3.585973, 9.637999, 23.736907, 56.547512, 132.884543 at L = 1..6 with log_3 step 0.777708, forcing a term-by-term cost r^1.277708 against a budget r^1 and an error no better than r^1.170497, worse than the exact floor Lambda_L >= 2^L - 1 gives; closing it needs |S_r| = O(r^0.222292), below the square-root floor. Witness: lab/circle-crop transform mass lines.
• [Conjecture] Phi = ind * Psi exactly with ind = prod_j (1 - p_j) (3^D/m)^L and Psi the dependence correction; carpet means settle at 0.942104 and 1.005714 while the global brackets [0.542697, 1.515753] and [0.793296, 1.374208] still widen with decelerating drift, so ind and Psi bounded is sufficient for the pointwise C(r) = Theta(r^(d-1)) and is the whole of what is left. Witness: lab/circle-crop digits and digittotal lines over eight carpet and five sponge windows.

The digit-restricted Mobius meter

• [Conjecture] That the defect x^(m/(2(q-m) ln q)) of a level-x^(alpha_q/2) distribution bound for the digit strings, a bound no page here states, is absorbed by the GRH saving at all: the cost-out sets two exponents from two unrelated statements on two yardsticks side by side and no derivation joins them, so it is neither a necessary condition nor a proof that a Type I estimate for M_F follows, the string-to-interval bookkeeping and the bilinear half of any such argument being untouched; the comparison is decided at the wall and nowhere else, lost there by a factor under 3 and won two steps later, so any sharper constant that moves q_0 must be re-costed rather than inherited. Witness: lab/mertens-numerology, mobius.md a power saving under GRH at large base.
• [Conjecture] Whether a Vaughan decomposition's coefficient sequence, a convolution and not a free sign vector, can be steered near the engineered sign vector that beats the Cauchy-Schwarz diagonal floor by a factor thirty-eight at a top box; and whether the arc regime M, N >= x^(2/5) carries a dyadic box with R = x^(alpha - o(1)), the middle-divisor question on which the balanced route's refutation for alpha < 2/5 is conditional. Witness: lab/rho-decoupling, sections menergy signed engineered and menergy type II.
• [Conjecture] A digit set satisfying condition (E) in one dimension whose sup-over-shift l^1 exponent obeys alpha_1 < 1/4 has Sum_{n in S_F, n <= x, gcd(n,q) = 1} mu(n) = O(A_F(x) (log x)^(-B)) for every B. The program is named: two Proved steps for mu, the rest set-only or coefficient-free, and the lattice branch of the source's Section 14 in general parameters owed. Base 10 fails on two independent numbers, 27/77 against 1/3 and 23/80 against 1/4. Witness: mobius.md The pair route. Superseded by the criterion row that names five lattice conditions and the threshold beta <= 1/4, under the same subsection in OPEN.
• [Conjecture] , whose owed list and whose base-10 diagnosis are both superseded. A digit set satisfying condition (E) in one dimension whose sup-over-shift l^1 exponent obeys alpha_1 < 1/4 has Sum_{n in S_F, n <= x, gcd(n,q) = 1} mu(n) = O(A_F(x) (log x)^(-B)) for every B. The program is named: the major-arc lemma and the level of distribution on an initial segment are Proved for mu, and the lattice branch is Proved in general parameters. Three things are owed and none is a new idea: the line branch at general base, whose two lemmas are set-free and coefficient-free but whose own conditions m_t < (2 - t) beta and N >= x^max((5/4) beta, (5 beta - 1/2)/3) are gathered into no statement yet; and the write-out at general base of two bookkeeping steps, the Parseval count of large frequencies for the Heath-Brown pieces and the dyadic reduction of the bilinear sum to the pair sum, both stated at source for arbitrary 1-bounded sequences. Base 10 now fails on one number only, the sharp threshold beta = inf_t m_t/(2 - t), at 23/80 against 1/4. Witness: mobius.md The pair route.

The digit-restricted Mobius exponent

• [Conjecture] theta(F) = 1/2 for every digit set with 2 <= |F| <= q - 1 and squarefree digit gcd - square-root cancellation against the set's own counting function: the 47 running-maximum exponents across q = 3, 4, 5, 10 read 0.4465..0.5358 with last-five-level drifts 0.0157..0.1056, while the full-set controls, whose limiting exponent is 1/2 under RH and at least 1/2 unconditionally, read 0.4413..0.4517 at the same depths; the finite tables are consistent and decide nothing, single-cut exponents scattering 0.22..0.53 on the same data. Witness: lab/mobius-designs, mobius.md.

Digit strings across divisors

• [Conjecture] The orbit-mean law at a pinned divisor: for F = {0..q-1} minus one digit and d = q^t - 1, the worst orbit-mean damping is k^(-1/t) (1 + o(1)), the orbit a q^j mod d carrying t - 1 undamped points and one damped by ~ 1/k; at q = 100, L = 12 the single-divisor probes read orbit mean 0.1059 at d = q^2 - 1 against k^(-1/2) and 0.2369 at d = q^3 - 1 against k^(-1/3), with the proper divisor d = 3367 | q^3 - 1 better at 0.0549 and d = 101 | q + 1 pinned but harmless at 0.0261, the kernel being flat across that whole orbit. Two values of t on one base with one dominant character are a check and not a law, and the o(1) is untested; t = 4 needs the orbit product analysed rather than counted, the exact count at d = q^4 - 1 being out of reach of the census. Witness: lab/rho-decoupling, mobius.md digit strings across divisors.

The repunit layer

• [Conjecture] The repunit drift is unbounded: X_k = (Z(R_k) - Phi_k)/Phi_k rises at every step from k = 8 and beats the random-lift limit sum_T 1/m_T = 1.41 at k = 13, so no pointwise Z(w) <= C w^(log 2/log 3) holds on the repunits and every exponent above log 2/log 3 survives; the blocking lemma is whether |union_T Occ_T| plus the deep tail is O(2^k). Witness: lab/ratio-set-saving (ratio.py repunit), coprime.md THE REPUNIT DRIFT.
• [Verified] The deepest first return of the critical band automaton grows below the critical sqrt(w) as a sign and not as an exclusion: log d_max on log w over 27 weights gives 0.4055, 95 percent [0.3293, 0.4818], but one deletion moves the slope to 0.4261 and the interval to 0.5034, covering 1/2, so the leave-one-out range [0.3829, 0.4261] is what stands; the median has no single exponent, 0.1802 on every weight against 0.2798 at Z >= 8 with 4 of 27 weights having Z <= 2; and a two-predictor fit puts 0.3385 on log w and 0.1313 on log Z, so controlling for sample size lowers the exponent and the drift below 1/2 is understated. Witness: lab/band-return-times critical.
• [Proved] The submasks of a binary K divisible by a divisor m of it are closed under complement in supp K, under disjoint union and under nested difference, so N_K(m) is even and every solution is a disjoint union of irreducible ones; the decomposition is not unique, so N_K(m) is the number of distinct unions of pairwise disjoint irreducibles and satisfies N_K(m) <= #packings <= 2^iota for the irreducible count iota, with N_K(m) = 2^iota if and only if the irreducibles are pairwise disjoint, and then they partition supp K. Depth-free. The antipodal and run families of the repunit lift are the equality case, which is why their counts are exact powers of two, and the converse fails, k = 7 with multiplier 19 having a power-of-two count and overlapping irreducibles. Witness: lab/band-return-times lift and check, the three closures asserted over every one of the 2^(k-1) sets T at every k = 2..9, and both witnesses pinned.
• [Verified] The depth-2 lift census of the repunit reads M_k = 2, 6, 14, 36, 68, 172, 306, 728, 1338, 2814, 5224, 11852, 20888, 43364, 84124, 172516, 327092 at k = 1..17 with M_k/2^k inside [1, 2.89356], and its Hankel matrix is 9 by 9 of full rank on all seventeen terms, so no linear recurrence of order at most 8; the irreducible supply Sum_T iota_T / 2^k sits inside [0.738281, 0.890625] at k = 2..12, the even readings falling from k = 6, while max_T iota_T grows 2, 3, 4, 5, 6, 10, 14, 24, 31, 50, 68, and the equality case holds for 1970 of the 2048 multipliers at k = 12 against 1986 whose count is a power of two. The sweep is exhaustive over every T inside [1, k-1]; the multipliers meet the residue classes 1 and 7 mod 9 and never 4, which is forced by a_T = Sum 3^i with i >= 1 and not a reading. Witness: lab/band-return-times lift.
• [Verified] A column transfer for the submask count of the lift half needs at least 253 states where the return half needs b/2: a machine reading the k columns with a state set free of k is a linear representation of the count as a series over the column word, so its state count is at least that series' Hankel rank, and the rank reads 3, 7, 14, 31, 62, 126, 253 at word length 1..7 on each side against the full 3, 7, 15, 31, 63, 127, 255, deficiency 0, 0, 1, 0, 1, 1, 2. The words reach length 14, so the floor holds at k <= 15 and is already worse than the 2^(k/2) meet in the middle; whether the rank is unbounded is observed and not proved, and a rank levelling off would be a poly-time machine, so the route is blocked and not closed. The irreducible count has the same full-rank Hankel to word length 5. Witness: lab/band-return-times lift.

Digit designs and the Euler product

• [Conjecture] The base-free Mertens rate of a two-digit design exceeds the design's mass rate. The maximum itself stays far under the mass at every level measured, max/2^L = 0.137890 at L = 18; it is the growth rate that runs ahead. The running maximum of sum nu* over degree below L reads 1, 1, 2, 3, 4, 7, 15, 23, 45, 86, 162, 331, 741, 1665, 3173, 7508, 17753, 36147 at L = 1..18; the ratio max/2^L bottoms at 0.079102 at L = 11 and rises in both residue classes of L mod 2 to 0.137890 at L = 18, with geometric mean steps 2.158562, 2.186288 and 2.127878 over the last 4, 6 and 8 levels against the mass rate 2. The measured log_2(max)/L is only 0.841199 at L = 18 and has not settled, so the limit is conjectural on an eighteen-level census; the step sits under q^(Re rho) = 2.207512 at base 3, a bracket between two different orders since base 3 leaves the base-free ladder at level 9 reading 38 against 45, and were the step to converge there then Re rho(q) = log lambda / log q for one base-free lambda, a family law for the locus of the design zeta's zeros. Witness: lab/carry-free-mobius verb exponent.

REFUTED

Automata

• [Refuted] "Fredkin's replicator is the mod-2 sum of the nine Moore cells": the named replicator B1357/S1357 is the eight-cell sum with kernel (1/x + 1 + x)(1/y + 1 + y) - 1, A160239, while the nine-cell sum is B1357/S02468, A246035, and the two agree at no generation past t = 0 except through the removed centre copy at t = 2^j. Witness: lab/life-census, A160239, A246035.

The tile monoid

• [Refuted] That every shape-distinct factorisation of a tile is axis-separable, which would have closed the question with a one-line lemma - [6]{(0,0),(2,2)} factors as c1 (x) c257.q3 and as c17.q3 (x) c1 with four irreducible letters and is not a rectangle, separability covering 121 of the 171 side-6 cross-shape tiles and none of the diagonal family, and the diagonal and antidiagonal families put a non-separable witness at every side with two distinct prime factors, so the door stays open. Witness: lab/code-factorisation, magic.md.
• [Refuted] That the mechanism of a shape-distinct factorisation is always a side-6 cross-shape collision sitting inside the word - the side-12 tile [12]{(0,0),(3,3)} has cut set {1,2,3,4,12} with no cut at 6, so no side-6 collision sits inside it, and its two readings differ in length; the earlier statement was read off a sample of words over plane codes and is a property of that universe, not a law. Witness: lab/code-factorisation, magic.md.
• [Refuted] That the length and the side multiset of a factorisation are invariants of the composite - true inside the magic-word submonoid, where every plane code has prime side and every word over it has length exactly the number of prime factors of the side, and false in the full tile monoid, first at side 12, where [12]{(0,0),(3,3)} reads at lengths 3 and 2; the plane-code census could not have found the witness, since the short reading needs the irreducible side-4 letter [4]{(0,0),(3,3)}, which is no plane code. Witness: lab/code-factorisation, magic.md.
• [Refuted] That the cut set L(C) is closed under lcm, and with it the naive reading that any two factorisations refine to a common one - the line {0,3} at N = 12 has L = {1,2,3,4,12}, holding 2 and 3 and not 6, and the first failure by mask order at that side is the line {1,2} with the same cut set; 132 failures over every line to N = 20 and 2376 over the 339795 side-12 plane-code composites, against zero failures of gcd closure in both sweeps. Witness: lab/code-factorisation, magic.md.
• [Refuted] That the commuting pairs are the four corner cells plus the scale-free families of row, column, full tile, diagonal and antidiagonal - the cells [3]{(1,1)} and [5]{(2,2)} commute at side 15, both readings giving [15]{(7,7)}, and neither is a corner cell nor a member of any of those families; the four-corner picture is an artifact of gcd(1,2) = 1 at sides (2,3), the correct criterion for one-cell letters being a(n - 1) = b(m - 1). Witness: lab/code-factorisation, magic.md.
• [Refuted] That the reachable literature cannot reach the tile factorisation question, an earlier positioning against graph products - the isomorphism-versus-equality gap is real for graph products, but Voet and De Novellis, arXiv:2510.25292, is a binary-matrix paper working under equality that already contains fixed-shape uniqueness, the prime vocabulary, the shape-distinct factorisation, the non-prime factor sizes and a decomposition graph enumerating every factorisation; the Proved core recorded above is a rediscovery, and the single verbatim quotation the old positioning rested on could not be recovered from its source and is withdrawn rather than carried. Witness: magic.md, arXiv:2510.25292.
• [Refuted] The annotation 1125899839733761 = 65535^2 in a draft of the prime-power counts - the integer is right and its name is wrong, 1125899839733761 = (2^25 - 1)^2 = 33554431^2 at side 25, while 65535^2 = 4294836225 is a side-16 term; transcription, not mathematics, and the generator now prints the identity beside the value. Witness: lab/code-factorisation.

The component exponent

• [Refuted] That a difference between the component growth rate and the frequency-average prediction is a non-stationary result - on the whole settled class the exponent is a function of the letter frequencies alone, so Thue-Morse returns exactly what a periodic word of the same frequencies and almost every Bernoulli word return, and the tree's own stationary controls miss the prediction by the same amount; the difference refutes the frequency functional, not stationarity, and an aperiodic word witnesses nothing here. Witness: connectivity.md, lab/magic-words.
• [Refuted] That the component exponent is a function of the letter frequencies for every word whose frequencies exist - over {3, 6} the frequency vector (1, 0) carries the constant word at rate 0, the diagonal-at-squares word at rate log 2, and the diagonal-at-powers-of-2 word with no rate at all; the statement holds only where both letters have positive frequency, and the same correction restores every family's closed-form rate. Witness: connectivity.md, lab/magic-words.
• [Refuted] That the top Lyapunov or matrix-norm exponent of the cocycle is the component exponent, and with it every route to chi through a norm theorem, a joint spectral radius or a projective contraction of forward orbits - along 3^inf the largest entry of M_3^L is exactly 2^(L+2) - 2, reading 6, 14, 62, 1022, 262142, 17179869182 at L = 1, 2, 4, 8, 16, 32, so the norm exponent is log 2, while comp(A_(3^L)) = 1 at every L and the component exponent is 0; the observation functional gamma is a fixed vector of both heavy matrices and so sits on the boundary of the dual cone, which is the geometric form of the same obstruction. Witness: connectivity.md, lab/magic-words.
• [Refuted] That entrywise positivity in the standard basis decides whether the letter matrices preserve a common cone - neither M_3 nor M_7 is a non-negative matrix, so the test never applied, and none of the 8190 products of length at most 12 is entrywise positive, yet a common invariant cone does exist in the chart normalised by the right eigenvector phi = (1,2,2,4)^T and the pair is primitive in it at length 3. Witness: connectivity.md, lab/magic-words.
• [Refuted] That the Thue-Morse word is a named word along which the component exponent fails to exist - it converges, exactly, to (1/2) log 6 on every gasket-against-domino pair with a two-sided certificate; the genuine non-existence witness is the gasket at the powers of 2 at the boundary frequency (1, 0), whose orbit closure is countable and not minimal, so no uniquely ergodic minimal word is implicated. Witness: connectivity.md, lab/magic-words.

Spin

• [Refuted] The corner ripple as a complete invariant of the transpose class - a design that is a solid segment has M(r) = c r exactly about a fixed point on it, so its ripple vanishes identically, and code 7, the solid row, and code 273, the solid diagonal, are two such designs of dimension exactly 1 in different transpose classes carrying the same zero ripple; the census reads them at swings 0.01114 and 0.01217 and mutual gap 0.01371, all discretisation, and no bar is needed for the conclusion. It does separate both named equal-mass pairs, at 2.9 and 8.2 times the drift bar. Witness: lab/spin-census.
• [Refuted] The Gaussian Farey counted by primitive representations in Z[i] modulo units - the norms below 2n^2 with a primitive representation run 2, 3, 6, 9, 13, 17, 23, 29, 35, 44 against new(n) = 2, 3, 9, 11, 22, 18, 40, 38, 55, 52, agreeing only at n = 1, 2; primitivity is the wrong condition, since (3,4) is primitive and 25 is a square, so sqrt(25)/5 = 1 is old at 5. The correct criterion is freedom from the squares of the primes of n. Witness: lab/spin-census.
• [Refuted] The disc and the box read the same Gaussian Farey - restricting to 0 <= a, b <= n instead of the disc of radius sqrt 2 breaks the criterion at n = 3, witness the radius 4/3: 16 is free of 9 and 4/3 < sqrt 2, but 16 = 4^2 + 0^2 needs a coordinate above 3, and the box counts 2, 3, 7, 9, 17, 14, 31, 27, 41, 38 part from the disc counts from n = 3 on. Witness: lab/spin-census.
• [Refuted] The coprime law survives the spin - flat layers at coprime odd scales are exactly uncorrelated, but their ring profiles over the inscribed disc correlate at +0.38 for (3, 5), -0.33 for (5, 7) and +0.38 for (9, 13), no better than gcd pairs; the cancellation is separable in x and y and the spin discards the angle. Witness: mrlylab test the_coprime_law_dies_under_the_spin.

Base-q design counts

• [Refuted] The base-q, D = 3 count 2, 22, 111618, 6005363762644688, 7089215977519836239803174210135872 has no OEIS entry - it is A398348, toroidal n x n x n binary arrays up to layer rotation, layer reflection and axis permutation, data verbatim with a b-file to n = 14 and A255016 as the two-dimensional case; the null search ran against a dump older than the entry. Witness: A398348, lab/oeis-terms.

Fill polynomials

• [Refuted] Seven D = 4 fill-polynomial remainders labelled irreducible over Q of degree 4 - palindromic signatures (1,0,1,0,1), (1,0,2,0,1), (1,1,2,1,1), (1,2,3,2,1), (1,3,2,3,1), (1,4,2,4,1), (1,4,5,4,1) - split over Q into two centered-polygonal quadratics k n^2 + k n + 1; rational-root peeling proves irreducibility only through degree 3, and no qualify/fail verdict moves since no factor is linear. Witness: lab/fill-polynomials.

Diagonal designs

• [Refuted] The diagonal designs are codes 98, 140 and 266 with Z_F(n) = 3^n - 2 - the identity fails at n = 0 (sides 0 and -1) and holds for every n >= 1, and the list is short by one: of the C(9,3) = 84 three-corner subsets of the 3 x 3 digit square, four are diagonal through n = 6, {(0,1),(1,0),(2,2)} (266), {(0,1),(1,2),(2,0)} (98), {(0,2),(1,0),(2,1)} (140) and {(0,2),(1,1),(2,0)}, code 84 (the value 148 once quoted for it names {(0,2),(1,1),(2,1)}, not a permutation design); the three named codes hold to n = 11 with every pair checked for collinearity at n = 7. Witness: gasket-ray-machine.

The Farey stack

• [Refuted] Stack brightness encodes the Mobius function, so a Baez-Duarte meter can replace the Franel table - neither brightness carries factorization data: the Farey stack gives B_Q(a/b) = floor(Q/b), 199 distinct values for the 10,000 denominators at Q = 10,000, every denominator from 5,001 to 10,000 sharing brightness 1 while mu runs -1, 0, +1; the gramstack gives 1 + K(a/b) with K = (-1)^a/b^2 for odd b and K = 0 for every even b, so all even denominators coincide; joining a node to a factorization through its denominator puts the arithmetic in the factorization. Witness: lab/carpet-stack-address.
• [Refuted] A design's Farey order is determined by its fill count - the stack of grid scales n = 1..Q lights exactly F_Q = {a/b : 1 <= a <= b <= Q, gcd(a,b) = 1} with brightness floor(Q/b), a boundary coordinate k/n reducing to a/b and recurring at every scale divisible by b, checked by literal stacking at Q = 30 on all 278 lit fractions; Farey order is Q, fill count plays no part, and every design gives the same sequence at fixed Q. Witness: lab/carpet-stack-address.

Integer census and avatars

• [Refuted] The D = 4 integer census has 350 qualifying signatures among 65536 designs - there are 12 distinct qualifying signatures, A000070(4), realized by 504 oriented designs of which 503 have k >= 2, meeting 33 full B_4 classes; 350 is none of 12, 504, 503, 402 or 33 and has no recoverable definition. Witness: divisor-avatars, A000070.
• [Refuted] The census lock predicate identifies 13 designs - it identifies 14: 9 on the P5 clause (62, 94, 110, 118, 122, 124, 188, 218, 230) and 5 on the edgeless clause (128, 134, 146, 148, 150), the recount to 13 dropping code 128, F = {111}, origin-free, edgeless, containing 111, of size 1. Witness: lab/fill-polynomials.

Component counts

• [Refuted] The boundary state of a Kronecker word grows with level, so the component count is not a finite-state function of the code sequence - one failed trial state (the four-corner partition of the running product, exact at length 2 and wrong on 20 of 216 words at length 3) bounds nothing; a linear representation of rank 4 exists and is exact on all 54240 words of length at most 4; what is unbounded is the naive geometric state kappa, which reaches 2^(L-1). Witness: order-sensitivity-of-kronecker-words.

The second moment of the rays

• [Refuted] The majorant Sum_z M_n(z)(M_n(z)-1) is a route to Conjecture W - it grows 2.907 a level at n = 13 against 2.573 for R itself, because it drops the coprimality of (s,t) and so counts each collinear pair once per common divisor; the exact identity R(n) = Sum_z P_n(z) survives and the golden ceiling M_n(z) <= F(n+1) - 1 survives, but the step from P_n(z) to M_n(z)^2 - M_n(z) does not. Witness: lab/gasket-witness-weights.

• [Refuted] The non-shift residual of the second moment converges as R(n)/phi^(2n) -> C ~ 3 at rate phi^2 - the normalised ratio reads 2.498, 3.045, 3.238, 3.182, 3.113, 2.983, 2.895 at n = 8, 10, 12, 13, 14, 15, 16, peaking at n = 12 and falling by 0.9724 per level, and R(n)/R(n-1) sits at 2.573, 2.561, 2.509, 2.541, a rate near 2.54, strictly below phi^2 = 2.618; what survives is R(n) = O(phi^(2n)) on n <= 16, lim R/phi^(2n) undecided, R = o(3^n) with room 3/2.54 rather than 3/2.618, and Conjecture Z untouched. Witness: lemma-b-pincer.

The slice ladder: rate and dead routes

• [Refuted] The decay rate of the slice-dimension excess is 3/4, 4/3 or 8/3 - at 320 digits over D = 2..100 the eigenvalue-scale one-step ratio extrapolates to 0.742874554813847413, residual 0.00712544518615 from 3/4, and r_inf = 1.34612251727283689, residual 0.0127891839395 from 4/3, both far outside the 4.5643e-8 parity split and the fit-order spread; on the dimension scale 2 r_inf = 2.6922450 against 8/3 = 2.666667; the coarse D <= 50 reading 0.373, inverse 2.68, and the sentence "the per-dimension factor approaches 3/4 from above" conflate the two scales; the constant is identified as prod_{k>=2} cos(2 pi/3^k) = 0.7428747134, within 1e-8 at D = 61. Witness: slice-recurrence-order.
• [Refuted] The excess has the clean shape dim_slice - (d-1) = (-1)^(D+1) C r^(-D) + o(r^(-D)) with a constant C - |delta_D| r_inf^D climbs from 52.4976468882 at D = 60 to 88.3872395676 at D = 100, a log-linear fit puts the prefactor at D^1, and the form is |delta_D| ~ A D r_inf^(-D) with A ~ 0.897520192686; the linear factor is the parity factor D - 1. Witness: slice-recurrence-order.
• [Refuted] Conjecture S stated against the threshold q^(D-1) - at base 3, sgn(rho_D - 3^(D-1)) matches the hypothesised sign in 25 of 49 cases over D = 2..50, is wrong already at D = 3 where the difference is -1.627718676730986, and exact real-root counting finds no eigenvalue above 3^(D-1) at any D = 2..20; at base 5, rho_D < 5^(D-1) at every tested D, rho_3 - 25 = -1.5341439003; the only threshold that carries the statement is fill/3, the eigenvalue-scale form of d - 1. Witness: slice-recurrence-order.
• [Refuted] The even carry block has exploitable matrix structure - over D = 2..20 the banded-plus-low-rank form does not exist (Toeplitz displacement rank equal to the full dimension from D >= 5, tridiagonal remainder of rank n-1 at odd and n at even D), total nonnegativity fails for every D >= 4 with an exact negative minor per row, only D = 2 is symmetric and only D = 3, 4 are positively diagonally symmetrizable (weights (1,6)), and M_even - (fill/3) I is Metzler rather than a Z-matrix, so the M-matrix route is circular; the matrices have n distinct real roots at every tested D, which is spectrally useless. Witness: slice-recurrence-order.
• [Refuted] A simple positive test vector certifies the Collatz-Wielandt bound - the all-ones vector has ratios mixed around fill/3 for every D = 3..50 (D = 2 excepted, that matrix being one by one), one-parameter cosine, alternating and centred-quadratic corrections succeed only at D = 2, 3, 4, and the Gaussian exp(-3 i^2 / D) and binomial-centre profiles only at D = 2; the Perron vector certifies at every D <= 50 (worst discrepancy 9.15e-46), is peaked at index 0 and monotone non-increasing rather than bell-shaped, and has no closed form. Witness: slice-sign-even-half.
• [Refuted] A cheap route proves the spectral separation rho_D/|lambda_2| -> 1 - the common-amplitude Gaussian kernel predicts a limiting ratio 9 where the truth is (D+2)/(D-2) -> 1, discarding an order-D parity modulation; the zero-shift 2x2 Schur complement has median relative error 0.625 over D = 2..40 and worst 0.9998, deteriorating with D; the Perron profile peaks at index 0 for all 39 tested D, a boundary-centred half-Gaussian at median R^2 = 0.99999, not near D/6; the second eigenvector has one sign change at every even D and at D = 3, 5 but several at every odd D from 7 to 39; separation holds numerically to D = 60, and a proof must be uniform in a margin of order 4/D. Witness: slice-recurrence-order.

Coprimality at dimension one

• [Refuted] The universal pair-prefix transfer matrix is a route to Conjecture W - its Perron root is k^2 = 4 for every D under coupled digit vectors, or 4^D under the scalar tensor reading, never 3; the 3 in W belongs to the shift multiplier-pair automata, where lambda(1, 3^r) = 3 exactly and every other coprime pair has lambda <= 2, with 2 attained at (1,4); the octave census that was fitted is the unweighted count, not W's weighted (3/2)^K sum, its exponent on the stabilised octaves j = 0..3 is 9.36, and the all-octave fit alpha = 2.956, CI [2.682, 3.257], leans on right-truncated high octaves with its constant drifting C = 1.042, 1.136, 1.244, 1.356 at n = 13..16, residual Durbin-Watson 0.261. Witness: gasket-ray-machine.
• [Refuted] Weil's theorem reaches the coprimality window - over F_3[t] the restricted coprime count grows like 4^n, a positive density among ordered pairs, so gamma = log_3(4) = 1.261860 and the window analogue (gamma/2, 1/2] is empty, gamma/2 = 0.630930 already above 1/2; the quantity is positive-density counting with no zeta error term, so the framework is sound over a field where the Riemann hypothesis is a theorem while carrying no zeta content; the like-for-like test is the F_q[t] analogue of the ray-multiplicity second moment, still undone. Witness: lab/function-field-density.

Conjecture S: the even-half transient

• [Refuted] The even half's early W_k sign alternation at base 3 persists, and a conjecture can be built on it - the alternation is a transient: over all even D the first break is D = 6, k = 4, then (8,6), (10,8), the rule k = D - 2 dying at D = 16 where the first break is k = 16, then 20, 24, 28, 34, 40 at D = 18..26 and none through k = 40 for D = 28..40 (inside the window even 12 <= D <= 22 the first break is (D,k) = (12,10)); structurally W_L ~ C 3^(L-1) rho^(L-1)(3 rho - fill) makes the eventual W-sign the even half itself; the pointwise route is dead too, V_2 < 0 for even D >= 6. Witness: slice-recurrence-order, slice-sign-even-half.

Conjecture S: even half

• [Refuted] The base-3 resummation mechanism built on those identities - the scaling limit 3^n f_n(psi)/fill^n -> Sigma(psi) is false, the exact tower falling geometrically to 0 as it must since a nonzero limit would contradict rho_D < fill/3; the claimed absolute convergence is false, the per-decade absolute mass of |G(m pi)|^(D-1) Ntilde(m pi) growing at D = 4 (block ratios 1.081 to 1.115 out to m = 2e7) and rising through 1 at D = 6, 8; the tail-to-lead figures -0.1812/-0.0464/-0.0121 are artifacts of the m <= 199 cutoff, still moving at m <= 2e5; Sigma(pi) > 0 is unproved at every D; the leader bound max_(m>1) |G(m pi)| = |G(7 pi)| = 0.2520527 holds to m <= 20001. Witness: slice-sign-even-half.
• [Refuted] V_(2j+1) < 0 for every even D >= 4 - at D = 4, V_L > 0 for every L <= 40 (V_1 = +4 exactly) and at D = 6, V_3 = +135092 > 0; the odd-L dip is a transient of length about 0.055 D^2, and D = 2, 4 never dip. Witness: slice-sign-even-half.
• [Refuted] Certificate depth K = 2 closes every even D at base 5 - 5 beta_3 < fill beta_2 holds for even 16 <= D <= 64 and fails at every even D = 66..320, first at D = 66 at the edge carry |c| = 32 = (D-2)/2, relative deficit -2.19e-43; beta_K has Fourier support 5^(-K) Z, the dominant frequency n = +-1 gives Sigma_K(c) ~ 2 T_K(1) cos(2 pi c/5^(K+1)), so depth K sees only carries inside the quarter-period |c| < 5^(K+1)/4 and dies at D = 2 ceil(5^(K+1)/4) + 2 - predicted deaths 16, 66, 316, 1566 at K = 1, 2, 3, 4, the first three exact - every fixed depth is finite, Theta(log D) growth is necessary, and the minimal K equals the row certificate's t(D) at every breakpoint tested (14|16, 64|66, 314|316). Witness: slice-sign-even-half.
• [Refuted] The base-3 certificate depth is exactly (9/160) D^2 - exact lower bounds put the residual at +2.00 by D = 120 and +9.78 at D = 178; 9/160 = 0.05625 is the first two digits of the true constant ln(R)/4 = 0.0568486146.... Witness: slice-sign-even-half.

The 2-adic Smith cascade

• [Refuted] The unified amplitude law max L_j = J(k - 1 - T(j-1)) with T triangular - fitted at j = 2, 3 where triangular and arithmetic indices coincide, it fails at j = 1 (true index k) and at j = 4, witnesses D = 689 and D = 1377. Witness: slice-sign-even-half.

The ratio-set power saving

• [Refuted] A uniform D_n(q) <= C 2^n / q on the binary base-3 multiples of q coprime to 3, the divisor input to the power saving - every binary m < 3^h makes m(1 + 3^h) binary below 3^(2h), so D_2h(1 + 3^h) >= 2^h while 4^h / q is only (2/3)^h of it, ratio (3/2)^h (1 + 3^(-h)) reading 2.0, 2.5, 3.5, 5.125, 7.625, 11.406, 17.094, 25.633 at h = 1..8, and at n = 20 the worst modulus below 500 is q = 244 = 1 + 3^5 at 1.8094; the breaking moduli are exactly the shift-ray weights. Witness: lab/ratio-set-saving.
• [Refuted] The short-witness route, bounding the count by 3^(level cap) - mean minimal witness length runs 3.875 to 27.287 and max 6 to 204 over height 32..16384, mean lev / log_3 height rises 1.553 to 3.305, and the share of occupied directions with lev <= 1.8073 log_3 x falls 0.875 to 0.3102, so every cap below 2 log_3 x loses a majority of the count. Witness: lab/ratio-set-saving.
• [Refuted] Two write-up claims of the first pass, caught by the adversarial read and corrected in place - the log Z_max / log W band was printed as [0.5000, 0.6404] when the script's own W = 128 row reads 0.70099, true band [0.5000, 0.7010] and margin 0.106 not 0.167; and ten Z_max values were listed against nine arguments, the duplicate Z_max = 30 at both W = 128 and W = 256 having been dropped, shifting every later argument onto the wrong weight. Both are transcription, not mathematics. Witness: lab/ratio-set-saving.
• [Refuted] beta >= log 2 / log 3 as a proved lower end of the weight-layer corridor - the binary-weight floor Z(w) >= #{coprime submasks} is proved, but at w = (3^k-1)/2 the coprime cut leaves 2, 6, 8, 30, 24, 126, 112 against w^(log 2 / log 3) = 2.4, 5.0, 10.3, 20.6, 41.3, 82.6, 165.3 for k = 2..8, beating the exponent at odd k and losing at even k, and no family supplies infinitely many good weights; the lower end is Conjecture. Witness: lab/ratio-set-saving.
• [Refuted] The metric route to the weight-layer saving: the sandwich Z(w) <= 2 N_P(1/w) is proved, but the cover of the slope set measures too large, n N_P(3^-n) / 3^n rising 2.4132 -> 2.4785, log_3 N_P / n rising 0.8783 -> 0.8997 and the step exponent rising 0.9333 -> 0.9504 over n = 12..18, every reading monotone and every one above the 0.8073 needed, so the covering route caps at O(w / log w) exactly like the congruence seed; a missing-digit rational-counting import belongs at the 3-adic ratio set R_inf and not at the slope variable. Witness: lab/ratio-set-saving.
• [Refuted] Three printed statements of the second pass, caught by the adversarial read and corrected in place - sigma_k < 1/9 was dated to k = 13 when it is 1/9 exactly at k = 2, 3, 4 and first below at k = 5, the bound already beating the trivial count at w = 13 (6 against 8.0) and w = 121 (46 against 73.3); the slope cover was computed on one swap half only, 51624 against the saturated 106994 at n = 12; and the backward moves were called one per residue class when 3k and 3k - z_1 are both 0 mod 3 and the class -z_2 mod 3 has no preimage. A fourth broke on the fix: the cover at n = 19 with three extra digits overflows int64 and printed a false 1.9533, so the generator now refuses past 1.5 * 3^(2n + extra) >= 2^63. Witness: lab/ratio-set-saving.
• [Refuted] The digit-congruence containment as printed: z_1 z_2^(-1) mod 3^k in R_k fails whenever 3^k | z_1, witness the occupied ray (9,1) at k = 2, where R_2 = {3} and the residue is 0; the true image is R_k union {0}, the counting bound's tail terms doubling to pay for the adjoined class, and the mod-3 dichotomy is the case k = 2 and not k = 1, R_1 being empty under u > 0. Witness: coprime.md, lab/occupancy-decay, lab/ratio-set-saving.
• [Refuted] The adversarial pass on the weight-layer run: an independent carry-pair dynamic programme reproduced every table to the last digit, the top-digit gap was checked over 14.3 million pairs at level 15 and found sharp and strict at minimum ratio 2.0000004 with extremal witness (3^14, (3^14 - 1)/2), Z(w) <= 2 |R_k| was checked at every weight to 8192 with no failure, the backward cone was rebuilt with zero mismatches, and three printed statements plus the |R_k| offset were broken and fixed in place. Witness: lab/ratio-set-saving.
• [Refuted] The adversarial pass on the run itself: the band was rederived on paper and sharpened to the halved interval -z_2/2 < j < z_1/2, an independent all-coprime-pairs automaton reproduced every A, Zsum and Z_max row to height 1024 and the full sweep, witnesses were reconstructed digit by digit for all 716 occupied directions to height 300 with zero failures, and Chow-Varju-Yu Theorem 1.2 and Kenyon were both verified accurate at source. Witness: lab/ratio-set-saving.